{"id":"n-2ee89feef605b3fb","revision":2,"first_index":0,"fingerprint":"f6-ea9a5170d08e2562","name":"basis_map_A007318 of Pascal-(1,9,1) array.","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":false,"rows":[["1"],["2","1"],["13","13","1"],["44","66","24","1"],["205","410","240","35","1"],["806","2015","1700","535","46","1"],["3457","10371","11265","5245","951","57","1"],["14168","49588","66336","41870","12376","1488","68","1"],["59449","237796","376984","298666","122386","24424","2146","79","1"],["246410","1108845","2035080","1948170","1031652","296010","42720","2925","90","1"],["1027861","5139305","10712025","12012270","7800942","2944962","620580","68595","3825","101","1"],["4273412","23503766","54858680","70585815","54432552","25639824","7234692","1168575","103380","4846","112","1"],["17797573","106785438","275883096","400548965","357769827","202425336","72199086","15774417","2027115","148406","5988","123","1"],["74055854","481363051","1363813644","2205981481","2241253922","1483271883","642694152","179377323","31276242","3297961","205004","7251","134","1"],["308289865","2158029055","6653007145","11863665155","13513358453","10249554155","5240279985","1793210325","402143595","57451853","5097515","274505","8635","145","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","27","135","891","5103","31347","185895","1121931","6711903","40330467","241805655","1451365371","8706597903","52244370387"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["9","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":"e-626d07f2a7c659cb","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["21","2"],["-29","-1"],["10","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["100","103","4"],["-40","-40","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["40","80","41","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161740,"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-a3fef8dca23d39bb"]},{"id":161741,"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-a3fef8dca23d39bb","e-d9851e23928f9acd"]},{"id":161742,"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-a3fef8dca23d39bb","e-d9851e23928f9acd"]},{"id":161743,"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-a3fef8dca23d39bb","e-d9851e23928f9acd"]},{"id":161744,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-d9851e23928f9acd"]}],"region_exact":{},"incoming_edges":[{"id":"e-a3fef8dca23d39bb","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-2ee89feef605b3fb","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":"1887b6866c509f3e29a3c9df2ee18fc4e84a63959a8cdb28d27509150bfdb5e4","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-7ac21ea1b3b7e282","role":"sequence","name":"Pascal-(1,9,1) array.","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-626d07f2a7c659cb","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-2ee89feef605b3fb","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":"8db3512e4df9b565947eacd497f53ef6f7b928f00cf75d31cde5b4398f23c755","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-d9851e23928f9acd","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2ee89feef605b3fb","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"1cdab1fe7fb663520419fc7321be61b9f82f329e12d32a3c3facc2a46cdd3601","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7ac21ea1b3b7e282","role":"sequence","name":"Pascal-(1,9,1) array.","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-c434a4d29001ca67","kind":"fingerprint_related","node_a_id":"n-2ee89feef605b3fb","node_b_id":"n-a793cd06bbb8aaa5","fingerprint":"f6-ea9a5170d08e2562","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ea9a5170d08e2562"},"other_id":"n-a793cd06bbb8aaa5","other":{"id":"n-a793cd06bbb8aaa5","name":"mobius_x_over_x_plus_one of Pascal-(1,9,1) array.","first_index":0,"fingerprint":"f6-ea9a5170d08e2562","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","13","13"]],"card_rows":[["1"],["1","2"],["1","13","13"],["1","24","66","44"]],"row_sums_preview":["1","3","27","135","891","5103","31347","185895"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7ac21ea1b3b7e282","steps":["x \u21a6 x/(x+1)"],"edges":["e-54eec0a056ff5a9d"],"text":"A143685 \u2192 x \u21a6 x/(x+1)"},"headline":"A143685 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-dff7f251a9fd5dc5","kind":"oeis_match","node_a_id":"n-2ee89feef605b3fb","node_b_id":null,"fingerprint":null,"oeis_id":"A080424","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A080424","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A080424","relation":"match_row_sums"}],"bfiles":[],"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,       1;\n  2:     13,      13,       1;\n  3:     44,      66,      24,       1;\n  4:    205,     410,     240,      35,       1;\n  5:    806,    2015,    1700,     535,      46,      1;\n  6:   3457,   10371,   11265,    5245,     951,     57,     1;\n  7:  14168,   49588,   66336,   41870,   12376,   1488,    68,    1;\n  8:  59449,  237796,  376984,  298666,  122386,  24424,  2146,   79,  1;\n  9: 246410, 1108845, 2035080, 1948170, 1031652, 296010, 42720, 2925, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9+9*x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(9 + 9*x)*P[n - 2, x] + (2 + 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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (9 + 9*x)*v[i - 2] + (2 + 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}}],"derivation":{"root":"n-7ac21ea1b3b7e282","steps":["basis A007318"],"edges":["e-a3fef8dca23d39bb"],"text":"A143685 \u2192 basis A007318"},"headline":"A143685 \u2192 basis A007318","chain":null}
