{"id":"n-9a3599832eeb87e6","revision":0,"first_index":0,"fingerprint":"f6-c163ccc412c10a39","name":"shift_x_plus_one of basis_map_A008288 of Triangle of coefficients in expansion of (1+2*x)^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":false,"rows":[["1"],["5","2"],["29","24","4"],["169","210","76","8"],["985","1632","920","208","16"],["5741","11890","9128","3232","528","32"],["33461","83160","80892","39280","10032","1280","64"],["195025","565474","666324","412488","144688","28768","3008","128"],["1136689","3766656","5214320","3930080","1761632","480768","78080","6912","256"],["6625109","24697890","39283856","34929280","19110240","6668480","1484544","203520","15616","512"],["38613965","159944280","287417908","294641952","190618528","81273856","23127680","4339456","514304","34816","1024"],["225058681","1025445938","2054507932","2386325960","1783743392","900432192","313166208","75057664","12156160","1268224","76800","2048"],["1311738121","6520076640","14410410120","18707165488","15873777648","9270043136","3819281664","1118594304","231216896","32915456","3065856","167936","4096"],["7645370045","41168607506","99498997944","142786503072","135645905392","90034868704","42943474432","14901403904","3762449152","682970624","86685696","7290880","364544","8192"],["44560482149","258405784728","677960945836","1065830669072","1120988991824","833978535168","452451388608","181815341312","54383474432","12052240384","1948820480","223080448","17100800","786432","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","57","463","3761","30551","248169","2015903","16375393","133019047","1080527769","8777241199","71298457361","579164900087","4704617658057"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179906,"position":0,"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-fc9cafc3d2e217ea"]},{"id":179907,"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-42c108d57a8a4978","e-fc9cafc3d2e217ea"]},{"id":179908,"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-42c108d57a8a4978","e-fc9cafc3d2e217ea"]},{"id":179909,"position":3,"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-42c108d57a8a4978"]},{"id":179910,"position":4,"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-42c108d57a8a4978"]}],"region_exact":{},"incoming_edges":[{"id":"e-42c108d57a8a4978","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9a3599832eeb87e6","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":"75bdf9ff9803140350279662fb2fc98187b9162b9b445414707edecf6cdc9027","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-33a50ee3b762dcdc","role":"sequence","name":"basis_map_A008288 of Triangle of coefficients in expansion of (1+2*x)^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]"}]},{"id":"e-fc9cafc3d2e217ea","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9a3599832eeb87e6","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":"5c20d9c9a44fb7419fa950f4e0ae390ed3265290f7f0de5249e495fe35bfed0f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-33a50ee3b762dcdc","role":"sequence","name":"basis_map_A008288 of Triangle of coefficients in expansion of (1+2*x)^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":[],"oeis":[],"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:       5,        2;\n  2:      29,       24,        4;\n  3:     169,      210,       76,        8;\n  4:     985,     1632,      920,      208,       16;\n  5:    5741,    11890,     9128,     3232,      528,      32;\n  6:   33461,    83160,    80892,    39280,    10032,    1280,      64;\n  7:  195025,   565474,   666324,   412488,   144688,   28768,    3008,    128;\n  8: 1136689,  3766656,  5214320,  3930080,  1761632,  480768,   78080,   6912,   256;\n  9: 6625109, 24697890, 39283856, 34929280, 19110240, 6668480, 1484544, 203520, 15616, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1-2*x)*P(n-2,x) + (6+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(1 - 2*x)*P[n - 2, x] + (6 + 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] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 - 2*x)*v[i - 2] + (6 + 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-71965e79e47a4526","steps":["basis A008288","basis A007318"],"edges":["e-9c90c90150c15b0e","e-42c108d57a8a4978"],"text":"A013609 \u2192 basis A008288 \u2192 basis A007318"},"headline":"A013609 \u2192 basis A008288 \u2192 basis A007318","chain":null}
