{"id":"n-5733d79f45a14084","revision":2,"first_index":0,"fingerprint":"f6-f06e51c139cfae05","name":"Triangle of coefficients in expansion of (2+3x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["2","3"],["4","12","9"],["8","36","54","27"],["16","96","216","216","81"],["32","240","720","1080","810","243"],["64","576","2160","4320","4860","2916","729"],["128","1344","6048","15120","22680","20412","10206","2187"],["256","3072","16128","48384","90720","108864","81648","34992","6561"],["512","6912","41472","145152","326592","489888","489888","314928","118098","19683"],["1024","15360","103680","414720","1088640","1959552","2449440","2099520","1180980","393660","59049"],["2048","33792","253440","1140480","3421440","7185024","10777536","11547360","8660520","4330260","1299078","177147"],["4096","73728","608256","3041280","10264320","24634368","43110144","55427328","51963120","34642080","15588936","4251528","531441"],["8192","159744","1437696","7907328","29652480","80061696","160123392","240185088","270208224","225173520","135104112","55269864","13817466","1594323"],["16384","344064","3354624","20127744","83026944","249080832","560431872","960740352","1260971712","1260971712","945728784","515852064","193444524","44641044","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","25","125","625","3125","15625","78125","390625","1953125","9765625","48828125","244140625","1220703125","6103515625"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A013620","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":"e-49cf64315b8edeba","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["2","5","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158312,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-0ecfea69e82c3845"]},{"id":158313,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-0ecfea69e82c3845","e-31e5a5a6b5727160","e-57079e0e5fed1718","e-cb393d3715764915","e-e9dcc990956c90e9"]},{"id":158314,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-0ecfea69e82c3845","e-31e5a5a6b5727160","e-57079e0e5fed1718","e-cb393d3715764915","e-e9dcc990956c90e9"]},{"id":158315,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-0ecfea69e82c3845"]},{"id":158316,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-31e5a5a6b5727160","e-cb393d3715764915","e-e9dcc990956c90e9"]},{"id":158317,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)","region_exact":null,"edges":["e-6f75aea89a6e01af"]},{"id":158318,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 2]","output_interval":"(-inf, 0]","statement":"an input root r <= 2 gives the root (r - a)/d <= 0; a larger root gives a positive root"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 2] to roots in (-inf, 0]: an input root r <= 2 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-6f75aea89a6e01af"]},{"id":158319,"position":7,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-57079e0e5fed1718"]}],"region_exact":{},"incoming_edges":[{"id":"e-e9dcc990956c90e9","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-5733d79f45a14084","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"b9ad6e3f732c781faf8ff443521084dc4b58293ad6cdb552534bc8b81c811db0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c8c19eba9d68881f","role":"sequence","name":"binomial","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-6f75aea89a6e01af","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-5733d79f45a14084","premise_count":0,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"4d5afddc0bf8bc6c59decb68c95ace54bc1ea439aa5c3c52ceebe13af8ee2116","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"claim":"basis_map_preserver","input_interval":"(-inf, 2]","output_interval":"(-inf, 0]","statement":"an input root r <= 2 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-49cf64315b8edeba","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-5733d79f45a14084","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"4e6f54402d338d2d229a540affaec25290cb8d9514e1bbaab7fd6121bba55321","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-31e5a5a6b5727160","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-5733d79f45a14084","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"ddc0c49ee94b349232c256700ca3d129bd38a02b8be2d075e14a13434abc3824","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-57079e0e5fed1718","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-5733d79f45a14084","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"46d8a4867d9190fbefbd127a9b693da6e4eda34b89c010eab9378df4913fc063","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-af754f07daf65b5e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-cb393d3715764915","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-5733d79f45a14084","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"ccb8f771a2091eea853f65915d93a56cfd9351d03f1532ae76fe6bd8874b5436","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-01cea33b6485d9cc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-0ecfea69e82c3845","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-5733d79f45a14084","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"14e77ccf4776e2059f159b41673304cc389545a03f1387ca914f560f6ecda433","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-1a3f71d8de8c31ec","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-7b068e1328e63a95","conclusion_name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+4x)^n."},{"id":"e-28556ceb40718bbf","operation":"basis_map","operation_detail":"A008288","conclusion_id":"n-f5630447d9b46bac","conclusion_name":"basis_map_A008288 of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-2ed5247925680563","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-23dbec3ecc58e8aa","conclusion_name":"basis_map_A008277 of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-4eef4a156684ec8f","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-a13c2de84911142b","conclusion_name":"basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-66f5a589f0412003","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-05c39cd3844b146d","conclusion_name":"basis_map_A007318 of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-b6ef91bfd675e934","operation":"basis_map","operation_detail":"A008288","conclusion_id":"n-f5630447d9b46bac","conclusion_name":"basis_map_A008288 of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-e341175dbafd44bc","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-c1936dc488503f26","conclusion_name":"basis_map_A013609 of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-cfb9e27fdcb193b1","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-1ce1c7eadcd8d479","conclusion_name":"obreschkoff_combination of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-cfb9e27fdcb193b1","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-1ce1c7eadcd8d479","conclusion_name":"obreschkoff_combination of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-1ea66849eac85f55","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-2eb61489a761e52c","conclusion_name":"one_plus_derivative of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-52ab624f3105e737","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-74c81a9d12bf4d45","conclusion_name":"derivative of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-7046ce3eaa519f49","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-eb2d732692093d3f","conclusion_name":"theta_plus_one of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-867523535b04b58e","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-f1a275eb676f791b","conclusion_name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-ad86b3979b25a235","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-05c39cd3844b146d","conclusion_name":"basis_map_A007318 of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-dc7bfe74658d97ca","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-01cea33b6485d9cc","conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j."},{"id":"e-e6c15cbf1d46e331","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-61a7d4345c34b084","conclusion_name":"theta_plus_one_squared of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-396a3b284da38893","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-ec6cfd69d11c8345","conclusion_name":"veronese_section_2_1 of Triangle of coefficients in expansion of (2+3x)^n."},{"id":"e-bdf59334244a08a5","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-f2968a7a3d3a43cd","conclusion_name":"veronese_section_2_0 of Triangle of coefficients in expansion of (2+3x)^n."}],"used_by_total":18,"links":[{"id":"l-163d45e06d5784dc","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-8e6dfc463e261fa6","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-8e6dfc463e261fa6","other":{"id":"n-8e6dfc463e261fa6","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","18"],["84","252","189"]],"card_rows":[["1"],["12","18"],["84","252","189"],["448","2016","3024","1512"]],"row_sums_preview":["1","30","525","7000","78750","787500","7218750","61875000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013610"],"edges":["e-97dd7dd90f768ab3"],"text":"A178822 \u2192 basis A013610"},"headline":"A178822 \u2192 basis A013610","chain":null}},{"id":"l-1cf75a4299172d4f","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-67e5bcee85c7fcd9","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-67e5bcee85c7fcd9","other":{"id":"n-67e5bcee85c7fcd9","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","1728","1296"],["13824","62208","93312","46656"],["331776","1990656","4478976","4478976","1679616"]],"card_rows":[["576","1728","1296"],["13824","62208","93312","46656"],["331776","1990656","4478976","4478976","1679616"],["7962624","59719680","179159040","268738560","201553920","60466176"]],"row_sums_preview":["3600","216000","12960000","777600000","46656000000","2799360000000","167961600000000","10077696000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013610"],"edges":["e-85e8843cd1dca083"],"text":"A038338 \u2192 basis A013610"},"headline":"A038338 \u2192 basis A013610","chain":null}},{"id":"l-20dcc1151139daf7","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-35f684e6e797bb02","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-35f684e6e797bb02","other":{"id":"n-35f684e6e797bb02","name":"basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["12","36","27"]],"card_rows":[["1"],["2","3"],["12","36","27"],["120","540","810","405"]],"row_sums_preview":["1","5","75","1875","65625","2953125","162421875","10557421875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013610"],"edges":["e-0adda7d4e180ead0"],"text":"A085881 \u2192 basis A013610"},"headline":"A085881 \u2192 basis A013610","chain":null}},{"id":"l-37ccdef07a984263","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-0f5fb19aff15410b","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-0f5fb19aff15410b","other":{"id":"n-0f5fb19aff15410b","name":"basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","6"],["8","24","18"]],"card_rows":[["2"],["4","6"],["8","24","18"],["16","72","108","54"]],"row_sums_preview":["2","10","50","250","1250","6250","31250","156250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013610"],"edges":["e-28b777b3c0b25332"],"text":"A028326 \u2192 basis A013610"},"headline":"A028326 \u2192 basis A013610","chain":null}},{"id":"l-3bd04805e53958b8","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-48d2b46df5963045","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-48d2b46df5963045","other":{"id":"n-48d2b46df5963045","name":"basis_map_A013610 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","33"],["484","1452","1089"]],"card_rows":[["1"],["22","33"],["484","1452","1089"],["10648","47916","71874","35937"]],"row_sums_preview":["1","55","3025","166375","9150625","503284375","27680640625","1522435234375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013610"],"edges":["e-bb91b382acf43386"],"text":"A038325 \u2192 basis A013610"},"headline":"A038325 \u2192 basis A013610","chain":null}},{"id":"l-3d622e70c4bd413e","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-01cea33b6485d9cc","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-01cea33b6485d9cc","other":{"id":"n-01cea33b6485d9cc","name":"Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["9","12","4"]],"card_rows":[["1"],["3","2"],["9","12","4"],["27","54","36","8"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038220"],"derivation":{"root":"n-01cea33b6485d9cc","steps":[],"edges":[],"text":"A038220"},"headline":"A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","chain":null}},{"id":"l-41dd78d990091443","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-48d0de0097720f76","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-48d0de0097720f76","other":{"id":"n-48d0de0097720f76","name":"basis_map_A013610 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-3"],["4","12","9"]],"card_rows":[["1"],["-2","-3"],["4","12","9"],["-8","-36","-54","-27"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013610"],"edges":["e-e876d74ed42231b4"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013610"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013610","chain":null}},{"id":"l-4b0978f21318a7db","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-408f7506d6594551","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-408f7506d6594551","other":{"id":"n-408f7506d6594551","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["36","48","16"]],"card_rows":[["1"],["6","4"],["36","48","16"],["216","432","288","64"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013611"],"edges":["e-0e13ca82ccc1a33c"],"text":"A038243 \u2192 basis A013611"},"headline":"A038243 \u2192 basis A013611","chain":null}},{"id":"l-52f13361e5e3735a","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-4f34a2344b135a41","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-4f34a2344b135a41","other":{"id":"n-4f34a2344b135a41","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["16","48","36"]],"card_rows":[["1"],["4","6"],["16","48","36"],["64","288","432","216"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013610"],"edges":["e-1a1a6d136d357a71"],"text":"A038208 \u2192 basis A013610"},"headline":"A038208 \u2192 basis A013610","chain":null}},{"id":"l-57c786551d8d3d13","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-f6495b2c898daf02","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-f6495b2c898daf02","other":{"id":"n-f6495b2c898daf02","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","6"],["27","36","12"]],"card_rows":[["3"],["9","6"],["27","36","12"],["81","162","108","24"]],"row_sums_preview":["3","15","75","375","1875","9375","46875","234375"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013609"],"edges":["e-618e7f7547136f14"],"text":"A120910 \u2192 basis A013609"},"headline":"A120910 \u2192 basis A013609","chain":null}},{"id":"l-7282fda34fd448cf","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-49350b617f7163d1","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-49350b617f7163d1","other":{"id":"n-49350b617f7163d1","name":"basis_map_A013610 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["24","72","54"]],"card_rows":[["1"],["4","6"],["24","72","54"],["160","720","1080","540"]],"row_sums_preview":["1","10","150","2500","43750","787500","14437500","268125000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013610"],"edges":["e-367cc697b56efa4f"],"text":"A356546 \u2192 basis A013610"},"headline":"A356546 \u2192 basis A013610","chain":null}},{"id":"l-829f1e90789c2039","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-fd2004a7ef60a2e8","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-fd2004a7ef60a2e8","other":{"id":"n-fd2004a7ef60a2e8","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["12","8"]],"card_rows":[["2"],["12","8"],["54","72","24"],["216","432","288","64"]],"row_sums_preview":["0","2","20","150","1000","6250","37500","218750"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["D"],"edges":["e-ae35042602575497"],"text":"A038220 \u2192 D"},"headline":"A038220 \u2192 D","chain":null}},{"id":"l-9645173e24311b06","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-cdd67e6ac4684e40","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-cdd67e6ac4684e40","other":{"id":"n-cdd67e6ac4684e40","name":"basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["8","24","18"]],"card_rows":[["1"],["2","3"],["8","24","18"],["40","180","270","135"]],"row_sums_preview":["1","5","50","625","8750","131250","2062500","33515625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013610"],"edges":["e-13e67b7b83b67a66"],"text":"A085880 \u2192 basis A013610"},"headline":"A085880 \u2192 basis A013610","chain":null}},{"id":"l-b22e5feeb600ba06","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-53e421fb0a6e1da4","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-53e421fb0a6e1da4","other":{"id":"n-53e421fb0a6e1da4","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","15"],["60","180","135"]],"card_rows":[["1"],["10","15"],["60","180","135"],["280","1260","1890","945"]],"row_sums_preview":["1","25","375","4375","43750","393750","3281250","25781250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610"],"edges":["e-0c010b1c5e20b0cb"],"text":"A178821 \u2192 basis A013610"},"headline":"A178821 \u2192 basis A013610","chain":null}},{"id":"l-b3b84b044aff93b3","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-9c0ed28aed848f75","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-9c0ed28aed848f75","other":{"id":"n-9c0ed28aed848f75","name":"basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["24","72","54"]],"card_rows":[["1"],["4","6"],["24","72","54"],["192","864","1296","648"]],"row_sums_preview":["1","10","150","3000","75000","2250000","78750000","3150000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013610"],"edges":["e-d66805981c716c2b"],"text":"A156992 \u2192 basis A013610"},"headline":"A156992 \u2192 basis A013610","chain":null}},{"id":"l-be96f38f852e6e9d","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-74c81a9d12bf4d45","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-74c81a9d12bf4d45","other":{"id":"n-74c81a9d12bf4d45","name":"derivative of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["12","18"]],"card_rows":[["3"],["12","18"],["36","108","81"],["96","432","648","324"]],"row_sums_preview":["0","3","30","225","1500","9375","56250","328125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["D"],"edges":["e-52ab624f3105e737"],"text":"A013620 \u2192 D"},"headline":"A013620 \u2192 D","chain":null}},{"id":"l-c33d56a81588e436","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-725ef1d8a9c0434d","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-725ef1d8a9c0434d","other":{"id":"n-725ef1d8a9c0434d","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["6","9"],["12","36","27"]],"card_rows":[["3"],["6","9"],["12","36","27"],["24","108","162","81"]],"row_sums_preview":["3","15","75","375","1875","9375","46875","234375"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["x \u21a6 x/(x+1)"],"edges":["e-8fbe952ff5bc6bbf"],"text":"A120910 \u2192 x \u21a6 x/(x+1)"},"headline":"A120910 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-ca27c65dd8f9e41b","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-545ecce3e4f3660f","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-545ecce3e4f3660f","other":{"id":"n-545ecce3e4f3660f","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["8","24","18"]],"card_rows":[["1"],["4","6"],["8","24","18"],["32","144","216","108"]],"row_sums_preview":["1","10","50","500","2500","25000","125000","1250000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["basis A013610"],"edges":["e-9d8a38d997eb16b9"],"text":"A135838 \u2192 basis A013610"},"headline":"A135838 \u2192 basis A013610","chain":null}},{"id":"l-cd606b97c184bee8","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-95f27714049d8d62","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-95f27714049d8d62","other":{"id":"n-95f27714049d8d62","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["40","120","90"]],"card_rows":[["1"],["8","12"],["40","120","90"],["160","720","1080","540"]],"row_sums_preview":["1","20","250","2500","21875","175000","1312500","9375000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013610"],"edges":["e-24fdbe8219e7135e"],"text":"A178820 \u2192 basis A013610"},"headline":"A178820 \u2192 basis A013610","chain":null}},{"id":"l-d15f8c38da8074b5","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-936114cbd4070c9a","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-936114cbd4070c9a","other":{"id":"n-936114cbd4070c9a","name":"basis_map_A013610 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","45"],["180","540","405"]],"card_rows":[["3"],["30","45"],["180","540","405"],["840","3780","5670","2835"]],"row_sums_preview":["3","75","1125","13125","131250","1181250","9843750","77343750"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A013610"],"edges":["e-127cb25e0d856e7f"],"text":"A344565 \u2192 basis A013610"},"headline":"A344565 \u2192 basis A013610","chain":null}},{"id":"l-e4d1043d20093747","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-c211a6e140527a22","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-c211a6e140527a22","other":{"id":"n-c211a6e140527a22","name":"basis_map_A013610 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["8","24","18"]],"card_rows":[["1"],["2","3"],["8","24","18"],["48","216","324","162"]],"row_sums_preview":["1","5","50","750","15000","375000","11250000","393750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013610"],"edges":["e-d8646d207d983420"],"text":"A196347 \u2192 basis A013610"},"headline":"A196347 \u2192 basis A013610","chain":null}},{"id":"l-e96fb8a7c8d83a12","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-9dc2047d1a16a51a","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-9dc2047d1a16a51a","other":{"id":"n-9dc2047d1a16a51a","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","9"],["36","108","81"]],"card_rows":[["1"],["6","9"],["36","108","81"],["216","972","1458","729"]],"row_sums_preview":["1","15","225","3375","50625","759375","11390625","170859375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013610"],"edges":["e-b88e4ed4246b1dea"],"text":"A038221 \u2192 basis A013610"},"headline":"A038221 \u2192 basis A013610","chain":null}},{"id":"l-eca6700721b1c51d","kind":"fingerprint_related","node_a_id":"n-5733d79f45a14084","node_b_id":"n-6874f381e9c7e0a2","fingerprint":"f6-f06e51c139cfae05","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f06e51c139cfae05"},"other_id":"n-6874f381e9c7e0a2","other":{"id":"n-6874f381e9c7e0a2","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f06e51c139cfae05","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","192","64"],["1728","3456","2304","512"],["20736","55296","55296","24576","4096"]],"card_rows":[["144","192","64"],["1728","3456","2304","512"],["20736","55296","55296","24576","4096"],["248832","829440","1105920","737280","245760","32768"]],"row_sums_preview":["400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":[],"edges":[],"text":"A038334"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","chain":null}},{"id":"l-0823ae34f3c7cd5e","kind":"oeis_match","node_a_id":"n-5733d79f45a14084","node_b_id":null,"fingerprint":null,"oeis_id":"A013620","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":46},"in_catalog":true,"oeis_id":"A013620","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-1577121c67eac57e","kind":"oeis_match","node_a_id":"n-5733d79f45a14084","node_b_id":null,"fingerprint":null,"oeis_id":"A074701","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A074701","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-81d8f8d918f99fb9","kind":"oeis_match","node_a_id":"n-5733d79f45a14084","node_b_id":null,"fingerprint":null,"oeis_id":"A000351","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000351","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a9bbfb50ad4b8491","kind":"oeis_match","node_a_id":"n-5733d79f45a14084","node_b_id":null,"fingerprint":null,"oeis_id":"A038220","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A038220","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ca1397c1d74299ec","kind":"oeis_match","node_a_id":"n-5733d79f45a14084","node_b_id":null,"fingerprint":null,"oeis_id":"A384565","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A384565","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A013620","relation":"identification"},{"oeis_id":"A038220","relation":"match_reversed_triangle"},{"oeis_id":"A000351","relation":"match_row_sums"},{"oeis_id":"A074701","relation":"match_row_sums"},{"oeis_id":"A384565","relation":"match_row_sums"},{"oeis_id":"A013620","relation":"match_triangle"}],"bfiles":[{"node_id":"n-5733d79f45a14084","oeis_id":"A000351","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":14900},{"node_id":"n-5733d79f45a14084","oeis_id":"A013620","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":46,"oeis_terms_agreeing":46,"oeis_terms_available":46,"term_count":990,"raw_size":21672},{"node_id":"n-5733d79f45a14084","oeis_id":"A038220","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":21672}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2,    3;\n  2:   4,   12,     9;\n  3:   8,   36,    54,     27;\n  4:  16,   96,   216,    216,     81;\n  5:  32,  240,   720,   1080,    810,    243;\n  6:  64,  576,  2160,   4320,   4860,   2916,    729;\n  7: 128, 1344,  6048,  15120,  22680,  20412,  10206,   2187;\n  8: 256, 3072, 16128,  48384,  90720, 108864,  81648,  34992,   6561;\n  9: 512, 6912, 41472, 145152, 326592, 489888, 489888, 314928, 118098, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A013620","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2,    3;\n  2:   4,   12,     9;\n  3:   8,   36,    54,     27;\n  4:  16,   96,   216,    216,     81;\n  5:  32,  240,   720,   1080,    810,    243;\n  6:  64,  576,  2160,   4320,   4860,   2916,    729;\n  7: 128, 1344,  6048,  15120,  22680,  20412,  10206,   2187;\n  8: 256, 3072, 16128,  48384,  90720, 108864,  81648,  34992,   6561;\n  9: 512, 6912, 41472, 145152, 326592, 489888, 489888, 314928, 118098, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + 3*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A038220","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     3,      2;\n  2:     9,     12,      4;\n  3:    27,     54,     36,      8;\n  4:    81,    216,    216,     96,     16;\n  5:   243,    810,   1080,    720,    240,     32;\n  6:   729,   2916,   4860,   4320,   2160,    576,     64;\n  7:  2187,  10206,  20412,  22680,  15120,   6048,   1344,   128;\n  8:  6561,  34992,  81648, 108864,  90720,  48384,  16128,  3072,  256;\n  9: 19683, 118098, 314928, 489888, 489888, 326592, 145152, 41472, 6912, 512;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + 3*x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-5733d79f45a14084","steps":[],"edges":[],"text":"A013620"},"headline":"A013620: Triangle of coefficients in expansion of (2+3x)^n.","chain":null}
