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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":"5dc5ce997ec965bcbcab20008ac4980199f863066eab27be80eb1563092a6e0a","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-5b7da9f56b3380cb","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.","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:         0;\n  1:         2;\n  2:        36,           8;\n  3:       388,         776,            24;\n  4:      3676,       24832,         11028,            64;\n  5:     33564,      566408,        849612,        134256,           160;\n  6:    303260,    11076368,      39082152,      22152736,       1516300,           384;\n  7:   2732156,   199663128,    1397326056,    1863101408,     499157820,      16392936,          896;\n  8:  24595932,  3435039776,   43219802892,  111786084480,   72033004820,   10305119328,    172171524,       2048;\n  9: 221378236, 57469421480, 1220768091732, 5485759477072, 6857199346340, 2441536183464, 201142975180, 1771025888, 4608;\n","recurrence_sentence":"The row polynomials satisfy (7*n-12)*P(n,x) = -(126*n-90+(196*n^2-672*n+380)*x+(343*n^3-1764*n^2+2681*n-1140)*x^2)*P(n-2,x) - ((98*n-70)*x+(343*n^2-931*n+490)*x^2-(343*n^2-833*n+420)*x^3)*d/dx P(n-2,x) + (77*n-118+(98*n^2-336*n+288)*x)*P(n-1,x) + ((49*n-84)*x-(49*n-84)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(126*n - 90 + (196*n^2 - 672*n + 380)*x + (343*n^3 - 1764*n^2 + 2681*n - 1140)*x^2)*P[n - 2, x] - ((98*n - 70)*x + (343*n^2 - 931*n + 490)*x^2 - (343*n^2 - 833*n + 420)*x^3)*D[P[n - 2, x], x] + (77*n - 118 + (98*n^2 - 336*n + 288)*x)*P[n - 1, x] + ((49*n - 84)*x - (49*n - 84)*x^2)*D[P[n - 1, x], x])/(7*n - 12)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(126*n - 90 + (196*n^2 - 672*n + 380)*x + (343*n^3 - 1764*n^2 + 2681*n - 1140)*x^2)*v[i - 2] - ((98*n - 70)*x + (343*n^2 - 931*n + 490)*x^2 - (343*n^2 - 833*n + 420)*x^3)*deriv(v[i - 2], x) + (77*n - 118 + (98*n^2 - 336*n + 288)*x)*v[i - 1] + ((49*n - 84)*x - (49*n - 84)*x^2)*deriv(v[i - 1], x))/(7*n - 12)); 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-5b7da9f56b3380cb","steps":["D"],"edges":["e-2b6a3c2ffd098beb"],"text":"A257617 \u2192 D"},"headline":"A257617 \u2192 D","chain":null}