{"id":"n-366b2525683554ec","revision":2,"first_index":0,"fingerprint":"f6-8ca1e631330e3f91","name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","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"],["8","1"],["64","16","1"],["512","192","24","1"],["4096","2048","384","32","1"],["32768","20480","5120","640","40","1"],["262144","196608","61440","10240","960","48","1"],["2097152","1835008","688128","143360","17920","1344","56","1"],["16777216","16777216","7340032","1835008","286720","28672","1792","64","1"],["134217728","150994944","75497472","22020096","4128768","516096","43008","2304","72","1"],["1073741824","1342177280","754974720","251658240","55050240","8257536","860160","61440","2880","80","1"],["8589934592","11811160064","7381975040","2768240640","692060160","121110528","15138816","1351680","84480","3520","88","1"],["68719476736","103079215104","70866960384","29527900160","8304721920","1660944384","242221056","25952256","2027520","112640","4224","96","1"],["549755813888","893353197568","670014898176","307090161664","95965675520","21592276992","3598712832","449839104","42172416","2928640","146432","4992","104","1"],["4398046511104","7696581394432","6253472382976","3126736191488","1074815565824","268703891456","50381979648","7197425664","787218432","65601536","4100096","186368","5824","112","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","81","729","6561","59049","531441","4782969","43046721","387420489","3486784401","31381059609","282429536481","2541865828329","22876792454961"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A038279","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158502,"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-b6a9810385e1026e"]},{"id":158503,"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-403b5075da2bf036","e-b6a9810385e1026e","e-e44a3cf5ca807197"]},{"id":158504,"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-403b5075da2bf036","e-b6a9810385e1026e","e-e44a3cf5ca807197"]},{"id":158505,"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-b6a9810385e1026e"]},{"id":158506,"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-403b5075da2bf036","e-e44a3cf5ca807197"]},{"id":158507,"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-e806b1e60392ec27"]},{"id":158508,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 8]","output_interval":"(-inf, 0]","statement":"an input root r <= 8 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, 8] to roots in (-inf, 0]: an input root r <= 8 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-e806b1e60392ec27"]}],"region_exact":{},"incoming_edges":[{"id":"e-403b5075da2bf036","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-366b2525683554ec","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":"6bbaf832bd3bed9f3ddaa3dfb493bb13951460f2345a4a6bae28d38b5fbaaba8","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-f211cb6153c37c6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(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-e806b1e60392ec27","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-366b2525683554ec","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":"3cec41f72a7cc0174bc34978b3cfc035534c4cf8c88eb104ab1ab95bd13c0c97","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, 8]","output_interval":"(-inf, 0]","statement":"an input root r <= 8 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-e44a3cf5ca807197","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-366b2525683554ec","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":"9089ae649a28402aca25c1242b43dc9fe2cce370dff0e843124b8313c339ae0e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-b6a9810385e1026e","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-366b2525683554ec","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":"f1b4cb56532596cb078409b0734c4def067831888b83c53df3cd7d408f795320","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-0da8e8e5bf86fd8d","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-20a6b63b1c7531be","conclusion_name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j."},{"id":"e-6eda6604dc872938","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-c5c52c7bc39f20ea","conclusion_name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j."},{"id":"e-a986b288b7f6003d","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-3dd184aba99388f7","conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j."},{"id":"e-b30dba0e9f8b339c","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-b8c0cc94a00a4dd8","conclusion_name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j."},{"id":"e-b4f43a8046f9cfa3","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-7a4a43fcd9fa4a51","conclusion_name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j."},{"id":"e-8868d67f2610da37","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-e3ab0b197a84a529","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j."},{"id":"e-8868d67f2610da37","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-e3ab0b197a84a529","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j."}],"used_by_total":7,"links":[{"id":"l-d4d0d2b66e8160ee","kind":"fingerprint_related","node_a_id":"n-366b2525683554ec","node_b_id":"n-326a4380cc213ab2","fingerprint":"f6-8ca1e631330e3f91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8ca1e631330e3f91"},"other_id":"n-326a4380cc213ab2","other":{"id":"n-326a4380cc213ab2","name":"basis_map_A013609 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-8ca1e631330e3f91","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","2"],["256","64","4"]],"card_rows":[["1"],["16","2"],["256","64","4"],["4096","1536","192","8"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013609"],"edges":["e-f16669f3777d4fa6"],"text":"A027467 \u2192 basis A013609"},"headline":"A027467 \u2192 basis A013609","chain":null}},{"id":"l-0450f9640548a0cc","kind":"oeis_match","node_a_id":"n-366b2525683554ec","node_b_id":null,"fingerprint":null,"oeis_id":"A001019","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":"A001019","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7348eb13aa551ef8","kind":"oeis_match","node_a_id":"n-366b2525683554ec","node_b_id":null,"fingerprint":null,"oeis_id":"A038279","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":41},"in_catalog":true,"oeis_id":"A038279","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-cb4efc843f07e7e6","kind":"oeis_match","node_a_id":"n-366b2525683554ec","node_b_id":null,"fingerprint":null,"oeis_id":"A013615","in_catalog":0,"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":false,"oeis_id":"A013615","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e3e7b0ddd44514e3","kind":"oeis_match","node_a_id":"n-366b2525683554ec","node_b_id":null,"fingerprint":null,"oeis_id":"A100062","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":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A100062","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A038279","relation":"identification"},{"oeis_id":"A013615","relation":"match_reversed_triangle"},{"oeis_id":"A001019","relation":"match_row_sums"},{"oeis_id":"A100062","relation":"match_row_sums"},{"oeis_id":"A038279","relation":"match_triangle"}],"bfiles":[{"node_id":"n-366b2525683554ec","oeis_id":"A001019","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":19978},{"node_id":"n-366b2525683554ec","oeis_id":"A013615","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":23471},{"node_id":"n-366b2525683554ec","oeis_id":"A038279","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":41,"oeis_terms_agreeing":41,"oeis_terms_available":41,"term_count":990,"raw_size":23471},{"node_id":"n-366b2525683554ec","oeis_id":"A100062","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":18,"oeis_terms_available":18,"term_count":200,"raw_size":20170}],"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:         8,         1;\n  2:        64,        16,        1;\n  3:       512,       192,       24,        1;\n  4:      4096,      2048,      384,       32,       1;\n  5:     32768,     20480,     5120,      640,      40,      1;\n  6:    262144,    196608,    61440,    10240,     960,     48,     1;\n  7:   2097152,   1835008,   688128,   143360,   17920,   1344,    56,    1;\n  8:  16777216,  16777216,  7340032,  1835008,  286720,  28672,  1792,   64,  1;\n  9: 134217728, 150994944, 75497472, 22020096, 4128768, 516096, 43008, 2304, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+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[(8 + 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] = (8 + 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":"A013615","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: 1,  8;\n  2: 1, 16,   64;\n  3: 1, 24,  192,   512;\n  4: 1, 32,  384,  2048,   4096;\n  5: 1, 40,  640,  5120,  20480,   32768;\n  6: 1, 48,  960, 10240,  61440,  196608,   262144;\n  7: 1, 56, 1344, 17920, 143360,  688128,  1835008,  2097152;\n  8: 1, 64, 1792, 28672, 286720, 1835008,  7340032, 16777216,  16777216;\n  9: 1, 72, 2304, 43008, 516096, 4128768, 22020096, 75497472, 150994944, 134217728;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (8+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[(8 + 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] = (8 + 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}},{"oeis_id":"A038279","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:         8,         1;\n  2:        64,        16,        1;\n  3:       512,       192,       24,        1;\n  4:      4096,      2048,      384,       32,       1;\n  5:     32768,     20480,     5120,      640,      40,      1;\n  6:    262144,    196608,    61440,    10240,     960,     48,     1;\n  7:   2097152,   1835008,   688128,   143360,   17920,   1344,    56,    1;\n  8:  16777216,  16777216,  7340032,  1835008,  286720,  28672,  1792,   64,  1;\n  9: 134217728, 150994944, 75497472, 22020096, 4128768, 516096, 43008, 2304, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+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[(8 + 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] = (8 + 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}}],"derivation":{"root":"n-366b2525683554ec","steps":[],"edges":[],"text":"A038279"},"headline":"A038279: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","chain":null}
