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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":"0a70482100d4ede8705007e133788144e1f3e3f036c577b6d0308dbfeacf6e18","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-817434b322e3be5b","role":"sequence","name":"basis_map_A013609 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:       1;\n  1:       7,        2;\n  2:      42,       28,        4;\n  3:     233,      258,       84,        8;\n  4:    1229,     1948,     1044,      224,       16;\n  5:    6264,    13064,     9992,     3504,      560,      32;\n  6:   31153,    81012,    81516,    40640,    10560,    1344,      64;\n  7:  152119,   475334,   597060,   390920,   143920,   29664,    3136,    128;\n  8:  732306,  2676940,  4047220,  3306240,  1595360,  464576,   79296,   7168,   256;\n  9: 3485585, 14607510, 25883640, 25456080, 15309840, 5832288, 1403136, 204288, 16128, 512;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3+52*n-32*n*x-16*n*x^2)*P(n,x) = (40*n^3+120*n^2+198*n+118+(16*n^3+48*n^2-60*n-92)*x-(88*n+88)*x^2-(16*n+16)*x^3)*P(n-1,x) + (52*n^2+52*n-39+(80*n^2+80*n-268)*x+(16*n^2+16*n-384)*x^2-144*x^3-16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((40*n^3 + 120*n^2 + 198*n + 118 + (16*n^3 + 48*n^2 - 60*n - 92)*x - (88*n + 88)*x^2 - (16*n + 16)*x^3)*P[n - 1, x] + (52*n^2 + 52*n - 39 + (80*n^2 + 80*n - 268)*x + (16*n^2 + 16*n - 384)*x^2 - 144*x^3 - 16*x^4)*D[P[n - 1, x], x])/(16*n^3 + 52*n - 32*n*x - 16*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((40*n^3 + 120*n^2 + 198*n + 118 + (16*n^3 + 48*n^2 - 60*n - 92)*x - (88*n + 88)*x^2 - (16*n + 16)*x^3)*v[i - 1] + (52*n^2 + 52*n - 39 + (80*n^2 + 80*n - 268)*x + (16*n^2 + 16*n - 384)*x^2 - 144*x^3 - 16*x^4)*deriv(v[i - 1], x))/(16*n^3 + 52*n - 32*n*x - 16*n*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-6466a8550c08a52f","steps":["basis A013609","D+1"],"edges":["e-0bad92788b2f6169","e-8a435babb742a662"],"text":"A116414 \u2192 basis A013609 \u2192 D+1"},"headline":"A116414 \u2192 basis A013609 \u2192 D+1","chain":null}