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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":"c03904a0fedc7df60dad0d78cc9ee648cda5d290e5137f5fbdc28336a4ba0227","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-d6be23d68d7706ed","role":"sequence","name":"basis_map_A013609 of A Riordan array of coordination sequences.","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:    12,      8;\n  3:    46,     72,     24;\n  4:   152,    384,    288,      64;\n  5:   466,   1632,   1968,     960,     160;\n  6:  1364,   6120,  10272,    8000,    2880,    384;\n  7:  3870,  21192,  45768,   49600,   28320,   8064,    896;\n  8: 10736,  69440, 183744,  256640,  203520,  91392,  21504,  2048;\n  9: 29282, 218496, 685152, 1174912, 1200960, 747264, 275968, 55296, 4608;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-6*n+4+(n^3-2*n^2-4*n+5)*x-(2*n^2-2)*x^2)*P(n,x) = -(2*n^3-16*n^2+28*n-8+(3*n^3-35*n^2+75*n-16)*x-(14*n^2-54*n+4)*x^2+12*n*x^3)*P(n-1,x) + (2*n^2-6*n+2+(8*n^2-28*n+8)*x+(9*n^2-43*n+9)*x^2+(2*n^2-24*n+2)*x^3-4*n*x^4)*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[(-(2*n^3 - 16*n^2 + 28*n - 8 + (3*n^3 - 35*n^2 + 75*n - 16)*x - (14*n^2 - 54*n + 4)*x^2 + 12*n*x^3)*P[n - 1, x] + (2*n^2 - 6*n + 2 + (8*n^2 - 28*n + 8)*x + (9*n^2 - 43*n + 9)*x^2 + (2*n^2 - 24*n + 2)*x^3 - 4*n*x^4)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 4 + (n^3 - 2*n^2 - 4*n + 5)*x - (2*n^2 - 2)*x^2)]];\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] = (-(2*n^3 - 16*n^2 + 28*n - 8 + (3*n^3 - 35*n^2 + 75*n - 16)*x - (14*n^2 - 54*n + 4)*x^2 + 12*n*x^3)*v[i - 1] + (2*n^2 - 6*n + 2 + (8*n^2 - 28*n + 8)*x + (9*n^2 - 43*n + 9)*x^2 + (2*n^2 - 24*n + 2)*x^3 - 4*n*x^4)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 4 + (n^3 - 2*n^2 - 4*n + 5)*x - (2*n^2 - 2)*x^2)); 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-ec35d1145e9a1211","steps":["basis A013609","D"],"edges":["e-7383986e8ecff3a4","e-d7a03c1dc37f2841"],"text":"A113413 \u2192 basis A013609 \u2192 D"},"headline":"A113413 \u2192 basis A013609 \u2192 D","chain":null}