{"id":"n-53e421fb0a6e1da4","revision":1,"first_index":0,"fingerprint":"f6-f06e51c139cfae05","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= 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"],["10","15"],["60","180","135"],["280","1260","1890","945"],["1120","6720","15120","15120","5670"],["4032","30240","90720","136080","102060","30618"],["13440","120960","453600","907200","1020600","612360","153090"],["42240","443520","1995840","4989600","7484400","6735960","3367980","721710"],["126720","1520640","7983360","23950080","44906400","53887680","40415760","17321040","3247695"],["366080","4942080","29652480","103783680","233513280","350269920","350269920","225173520","84440070","14073345"],["1025024","15375360","103783680","415134720","1089728640","1961511552","2451889440","2101619520","1182160980","394053660","59108049"],["2795520","46126080","345945600","1556755200","4670265600","9807557760","14711336640","15762146400","11821609800","5910804900","1773241470","241805655"],["7454720","134184960","1107025920","5535129600","18681062400","44834549760","78460462080","100877736960","94572878400","63048585600","28371863520","7737780960","967222620"],["19496960","380190720","3421716480","18819440640","70572902400","190546836480","381093672960","571640509440","643095573120","535912977600","321547786560","131542276320","32885569080","3794488740"],["50135040","1052835840","10265149440","61590896640","254062448640","762187345920","1714921528320","2939865477120","3858573438720","3858573438720","2893930079040","1578507315840","591940243440","136601594640","14635885140"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","25","375","4375","43750","393750","3281250","25781250","193359375","1396484375","9775390625","66650390625","444335937500","2905273437500","18676757812500"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","12"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167284,"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":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-0c010b1c5e20b0cb"]},{"id":167285,"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-0c010b1c5e20b0cb"]},{"id":167286,"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-0c010b1c5e20b0cb"]},{"id":167287,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-0c010b1c5e20b0cb"]}],"region_exact":{},"incoming_edges":[{"id":"e-0c010b1c5e20b0cb","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-53e421fb0a6e1da4","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":"0d1048b2b656c57966367e991f5297909fd1c99050bbf24457dc470022f220fe","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-79986ab5023c2764","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","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]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-02d6d20ef5a4902c","kind":"fingerprint_related","node_a_id":"n-74c81a9d12bf4d45","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-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-0ec6a19c778ce150","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-239ad4138bb031dd","kind":"fingerprint_related","node_a_id":"n-01cea33b6485d9cc","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-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-3330c863072c24fa","kind":"fingerprint_related","node_a_id":"n-0f5fb19aff15410b","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-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-45dae23b0e402b8a","kind":"fingerprint_related","node_a_id":"n-545ecce3e4f3660f","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-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-54db6632b61cd0cc","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-5e199a9406437c75","kind":"fingerprint_related","node_a_id":"n-4f34a2344b135a41","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-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-5efd7f1cd5654208","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-600561dda0142f1c","kind":"fingerprint_related","node_a_id":"n-6874f381e9c7e0a2","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-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-6c08215b1c9ba3ad","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-802f3a4ccc82c7e2","kind":"fingerprint_related","node_a_id":"n-48d0de0097720f76","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-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-8195549a449c881f","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-960b20777f9e4e0a","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-9cc24d57eb2d5137","kind":"fingerprint_related","node_a_id":"n-95f27714049d8d62","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-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-a2ebf95cc6d89bb4","kind":"fingerprint_related","node_a_id":"n-fd2004a7ef60a2e8","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-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-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-5733d79f45a14084","other":{"id":"n-5733d79f45a14084","name":"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":[["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)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013620"],"derivation":{"root":"n-5733d79f45a14084","steps":[],"edges":[],"text":"A013620"},"headline":"A013620: Triangle of coefficients in expansion of (2+3x)^n.","chain":null}},{"id":"l-cac9d6075e3ff1b4","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-d0c5b2e0d374c415","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-d45453ec0748b4fc","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-e0dae50359643d4b","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-f1990e8aaf4c47b8","kind":"fingerprint_related","node_a_id":"n-53e421fb0a6e1da4","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-f743a4d6b4b96f7d","kind":"fingerprint_related","node_a_id":"n-725ef1d8a9c0434d","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-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-fdbe1499c9b8fa26","kind":"fingerprint_related","node_a_id":"n-f6495b2c898daf02","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-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-45cb2f035b4888a8","kind":"oeis_match","node_a_id":"n-53e421fb0a6e1da4","node_b_id":null,"fingerprint":null,"oeis_id":"A036071","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":"A036071","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A036071","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-53e421fb0a6e1da4","oeis_id":"A036071","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":16134}],"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:     10,      15;\n  2:     60,     180,      135;\n  3:    280,    1260,     1890,       945;\n  4:   1120,    6720,    15120,     15120,      5670;\n  5:   4032,   30240,    90720,    136080,    102060,     30618;\n  6:  13440,  120960,   453600,    907200,   1020600,    612360,    153090;\n  7:  42240,  443520,  1995840,   4989600,   7484400,   6735960,   3367980,    721710;\n  8: 126720, 1520640,  7983360,  23950080,  44906400,  53887680,  40415760,  17321040,  3247695;\n  9: 366080, 4942080, 29652480, 103783680, 233513280, 350269920, 350269920, 225173520, 84440070, 14073345;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (2*n+8+(3*n+12)*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[Together[((2*n + 8 + (3*n + 12)*x)*P[n - 1, x])/(n)]];\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*n + 8 + (3*n + 12)*x)*v[i - 1])/(n)); 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}}],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610"],"edges":["e-0c010b1c5e20b0cb"],"text":"A178821 \u2192 basis A013610"},"headline":"A178821 \u2192 basis A013610","chain":null}