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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":"2082dc389a2555e84f4f5264ba2ded03743f9809d5f754a7c863741e967ae1d9","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-cffba19abf0eaf93","role":"sequence","name":"basis_map_A013611 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:     4;\n  2:    24,     32;\n  3:    92,    288,     192;\n  4:   304,   1536,    2304,     1024;\n  5:   932,   6528,   15744,    15360,     5120;\n  6:  2728,  24480,   82176,   128000,    92160,    24576;\n  7:  7740,  84768,  366144,   793600,   906240,   516096,   114688;\n  8: 21472, 277760, 1469952,  4106240,  6512640,  5849088,  2752512,   524288;\n  9: 58564, 873984, 5481216, 18798592, 38430720, 47824896, 35323904, 14155776, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-6*n+4+(2*n^3-4*n^2-8*n+10)*x-(8*n^2-8)*x^2)*P(n,x) = -(2*n^3-16*n^2+28*n-8+(6*n^3-70*n^2+150*n-32)*x-(56*n^2-216*n+16)*x^2+96*n*x^3)*P(n-1,x) + (n^2-3*n+1+(8*n^2-28*n+8)*x+(18*n^2-86*n+18)*x^2+(8*n^2-96*n+8)*x^3-32*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 16*n^2 + 28*n - 8 + (6*n^3 - 70*n^2 + 150*n - 32)*x - (56*n^2 - 216*n + 16)*x^2 + 96*n*x^3)*P[n - 1, x] + (n^2 - 3*n + 1 + (8*n^2 - 28*n + 8)*x + (18*n^2 - 86*n + 18)*x^2 + (8*n^2 - 96*n + 8)*x^3 - 32*n*x^4)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 4 + (2*n^3 - 4*n^2 - 8*n + 10)*x - (8*n^2 - 8)*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 16*n^2 + 28*n - 8 + (6*n^3 - 70*n^2 + 150*n - 32)*x - (56*n^2 - 216*n + 16)*x^2 + 96*n*x^3)*v[i - 1] + (n^2 - 3*n + 1 + (8*n^2 - 28*n + 8)*x + (18*n^2 - 86*n + 18)*x^2 + (8*n^2 - 96*n + 8)*x^3 - 32*n*x^4)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 4 + (2*n^3 - 4*n^2 - 8*n + 10)*x - (8*n^2 - 8)*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 A013611","D"],"edges":["e-14cc72d2ba3c3aa2","e-0c4b237fceb5e7c5"],"text":"A113413 \u2192 basis A013611 \u2192 D"},"headline":"A113413 \u2192 basis A013611 \u2192 D","chain":null}