{"id":"n-2a830d2c1dad85e9","revision":0,"first_index":0,"fingerprint":"f6-54d1a54334315095","name":"shift_x_plus_one of basis_map_A013610 of Square the entries of Pascal's triangle.","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","3"],["33","36","9"],["245","387","189","27"],["1921","3984","2916","864","81"],["15525","39915","39060","17820","3645","243"],["127905","392580","481545","298080","95985","14580","729"],["1067925","3810555","5621805","4402755","1947645","474579","56133","2187"],["9004545","36624960","63156240","59875200","33781860","11430720","2204496","209952","6561"],["76499525","349356915","689530320","767718000","527344020","229206348","61970832","9762768","767637","19683"],["653808673","3312402180","7364045745","9418861440","7630583940","4058477136","1418531940","315977760","41629545","2755620","59049"],["5614995765","31252739859","77279928165","111660284835","104262877620","65656524780","28212221268","8171200620","1533994605","172127835","9743085","177147"],["48416454529","293676488208","799500630852","1287897636960","1362115639665","991627230528","506272059216","181032581664","44425219905","7153589520","693707652","34012224","531441"],["418895174885","2750204625051","8173851682932","14524821598716","17163964090845","14187775368195","8397456127632","3579403308624","1088694266517","230282144235","32256106740","2735858268","117448461","1594323"],["3634723102113","25679895290532","82736425328193","160771827277152","209954896504353","194299378349076","130873160667249","64838895222816","23579442015129","6204217255236","1146755258649","141343465536","10591087689","401769396","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","78","848","9766","116208","1411404","17383584","216294534","2712176048","34217133028","433826815584","5522845782364","70550459395424","903871957376088"],"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"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-8","-6"],["13","9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-8","-14","-6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-498b697045e9102b","recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","3"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["9","18","9"],["-9","-18","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-5","-3"],["10","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182983,"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-e603f550182e4fa3"]},{"id":182984,"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-e603f550182e4fa3","e-e72bb21ada8e5030"]},{"id":182985,"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-e603f550182e4fa3","e-e72bb21ada8e5030"]},{"id":182986,"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-e603f550182e4fa3","e-e72bb21ada8e5030"]},{"id":182987,"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-e72bb21ada8e5030"]}],"region_exact":{},"incoming_edges":[{"id":"e-e72bb21ada8e5030","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-2a830d2c1dad85e9","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":"e1348b9395e3d7dbb97dd9a057ab93126516abfc5b280a3df500883081b6348b","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-117b55f7953e214b","role":"sequence","name":"basis_map_A013610 of Square the entries of Pascal's triangle.","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-498b697045e9102b","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-2a830d2c1dad85e9","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":"b895acdd8a81400c1b6ca5f6f8203dbfe02c139559a049516b73b29b63aec8de","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-e603f550182e4fa3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2a830d2c1dad85e9","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":"47c0c5b366aa11ef2c0d41c08148d2b7ea53302b8c28c1b6b983a4f81fc71690","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-117b55f7953e214b","role":"sequence","name":"basis_map_A013610 of Square the entries of Pascal's triangle.","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,         3;\n  2:       33,        36,         9;\n  3:      245,       387,       189,        27;\n  4:     1921,      3984,      2916,       864,        81;\n  5:    15525,     39915,     39060,     17820,      3645,       243;\n  6:   127905,    392580,    481545,    298080,     95985,     14580,      729;\n  7:  1067925,   3810555,   5621805,   4402755,   1947645,    474579,    56133,    2187;\n  8:  9004545,  36624960,  63156240,  59875200,  33781860,  11430720,  2204496,  209952,   6561;\n  9: 76499525, 349356915, 689530320, 767718000, 527344020, 229206348, 61970832, 9762768, 767637, 19683;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (13*n-8+(9*n-6)*x)*P(n-1,x) - (8+14*x+6*x^2)*d/dx 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[((13*n - 8 + (9*n - 6)*x)*P[n - 1, x] - (8 + 14*x + 6*x^2)*D[P[n - 1, x], 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] = ((13*n - 8 + (9*n - 6)*x)*v[i - 1] - (8 + 14*x + 6*x^2)*deriv(v[i - 1], x))/(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-b13531ad79e7c629","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-0e684cdba462224e","e-e603f550182e4fa3"],"text":"A008459 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A008459 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}
