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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":"63142589a13a841d6dd8fdf25db9c9bb9e5f89e093e711a14c971f84c14b738b","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-53dc6d1eeb1907f4","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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:             828,              896,               196;\n  1:           19440,            34776,             18816,              2744;\n  2:          431568,          1088640,            973728,            351232,             38416;\n  3:         9237888,         30209760,          38102400,          22720320,           6146560,            537824;\n  4:       192735936,        775982592,        1268809920,        1066867200,         477126720,         103262208,           7529536;\n  5:      3945417984,      18888121728,       38023147008,       41447790720,       26138246400,        9351683712,        1686616064,         105413504;\n  6:     79588604160,     441886814208,     1057734816768,     1419530821632,     1160538140160,      585496719360,      174564762624,       26985857024,       1475789056;\n  7:   1586874322944,   10028164124160,    27838869295104,    44424862304256,    44715220881408,    29245561132032,    12295431106560,     3142165727232,     425027248128,     20661046784;\n  8:  31340767878144,  222162405212160,   701971488691200,  1299147233771520,  1554870180648960,  1252026184679424,   682396426414080,   245908622131200,   54987900226560,   6611534970880,    289254654976;\n  9: 614120362979328, 4826478253234176, 17106505201336320, 36034536419481600, 50017168500203520, 47890001563987968, 32135338740105216, 15012721381109760, 4733740976025600, 940904070543360, 101817638551552, 4049565169664;\n","recurrence_sentence":"The row polynomials satisfy (7*n+16+7*x)*P(n,x) = (126*n+414+(98*n+448)*x+98*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 828 + 896*x + 196*x^2.","mathematica":"P[0, x_] := 828 + 896*x + 196*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((126*n + 414 + (98*n + 448)*x + 98*x^2)*P[n - 1, x])/(7*n + 16 + 7*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 828 + 896*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((126*n + 414 + (98*n + 448)*x + 98*x^2)*v[i - 1])/(7*n + 16 + 7*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-7dcf11eddf2b9a8a","steps":["basis A013609","D+1"],"edges":["e-322a2e4c0bbd3231","e-a807dd1bacb88ce8"],"text":"A038321 \u2192 basis A013609 \u2192 D+1"},"headline":"A038321 \u2192 basis A013609 \u2192 D+1","chain":null}