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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"7b50d11bac7b2110afe94c1fb104922aec71834abd3c4f680ae6f9e856ef5046","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-40f57961a3f2ab5d","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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]"}]}],"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:           649,            1680,              576;\n  1:         10043,           46728,            60480,            13824;\n  2:        142417,          964128,          2242944,          1935360,            331776;\n  3:       1917971,        17090040,         57847680,         89717760,          58060800,           7962624;\n  4:      24962905,       276187824,       1230482880,       2776688640,        3229839360,        1672151040,          191102976;\n  5:     317109419,      4193768040,      23199777216,      68907041280,      116620922880,      108522602496,        46820229120,         4586471424;\n  6:    3955895713,     60885008448,     402601731840,    1484785741824,     3307537981440,     4478243438592,      3472723279872,      1284211998720,       110075314176;\n  7:   48659465987,    854473474008,    6575580912384,   28987324692480,    80178430058496,   142885640798208,    161216763789312,    107158318350336,     34673723965440,     2641807540224;\n  8:  591844870441,  11678271836880,  102536816880960,  526046472990720,  1739239481548800,  3848564642807808,   5715425631928320,   5527431901347840,   3214749550510080,   924632639078400,    63403380965376;\n  9: 7132791765275, 156247045796424, 1541531882468160, 9023239885524480, 34719067217387520, 91831844625776640, 169336844283543552, 215553195261296640, 182405252744478720, 94299320148295680, 24410301671669760, 1521681143169024;\n","recurrence_sentence":"The row polynomials satisfy (24*n+35+24*x)*P(n,x) = (264*n+649+(576*n+1680)*x+576*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 649 + 1680*x + 576*x^2.","mathematica":"P[0, x_] := 649 + 1680*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((264*n + 649 + (576*n + 1680)*x + 576*x^2)*P[n - 1, x])/(24*n + 35 + 24*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 649 + 1680*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((264*n + 649 + (576*n + 1680)*x + 576*x^2)*v[i - 1])/(24*n + 35 + 24*x)); 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-26633b980c3508e8","steps":["basis A013610","D+1"],"edges":["e-edadd00b11ba8928","e-c19f8b0b8b3ff198"],"text":"A038226 \u2192 basis A013610 \u2192 D+1"},"headline":"A038226 \u2192 basis A013610 \u2192 D+1","chain":null}