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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":"84db9553368532e74c886f0362fa87b3607978f76113099cc8a3227d1524142e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7d781fe2b71cdf22","role":"sequence","name":"Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].","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:         1;\n  2:         6,         2;\n  3:        45,        30,         3;\n  4:       420,       420,        84,         4;\n  5:      4725,      6300,      1890,       180,        5;\n  6:     62370,    103950,     41580,      5940,      330,       6;\n  7:    945945,   1891890,    945945,    180180,    15015,     546,     7;\n  8:  16216200,  37837800,  22702680,   5405400,   600600,   32760,   840,    8;\n  9: 310134825, 827026200, 578918340, 165405240, 22972950, 1670760, 64260, 1224, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n^2-n+n*x)*P(n-1,x) + 2*n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - n + n*x)*P[n - 1, x] + 2*n*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 - n + n*x)*v[i - 1] + 2*n*x*deriv(v[i - 1], x))/(n - 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-7d781fe2b71cdf22","steps":["D"],"edges":["e-ee7f39ff4ff1008b"],"text":"A176230 \u2192 D"},"headline":"A176230 \u2192 D","chain":null}