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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":"aeb32197300b13abc5e2c4da6ef4eee03d2a99900884ce37fa3dcd219c7b9fe3","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-edcd820194369f8f","role":"sequence","name":"basis_map_A013611 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:       9,        4;\n  2:      62,       72,        16;\n  3:     371,      756,       432,        64;\n  4:    2049,     6152,      6096,      2304,       256;\n  5:   10768,    43248,     62816,     40832,     11520,      1024;\n  6:   54733,   276888,    537264,    509440,    245760,     55296,      4096;\n  7:  271653,  1662268,   4064880,   5136960,   3600640,   1379328,    258048,    16384;\n  8: 1324406,  9523560,  28208080,  44887040,  41832960,  23209984,   7368704,  1179648,   65536;\n  9: 6367895, 52665420, 183618720, 353907840, 414785280, 305335296, 140034048, 37945344, 5308416, 262144;\n","recurrence_sentence":"The row polynomials satisfy (128*n^3+248*n+64*n*x-128*n*x^2)*P(n,x) = (320*n^3+960*n^2+1372*n+732+(256*n^3+768*n^2+784*n+272)*x-(448*n+448)*x^2-(256*n+256)*x^3)*P(n-1,x) + (208*n^2+208*n+91+(640*n^2+640*n-32)*x+(256*n^2+256*n-1056)*x^2-1024*x^3-256*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[((320*n^3 + 960*n^2 + 1372*n + 732 + (256*n^3 + 768*n^2 + 784*n + 272)*x - (448*n + 448)*x^2 - (256*n + 256)*x^3)*P[n - 1, x] + (208*n^2 + 208*n + 91 + (640*n^2 + 640*n - 32)*x + (256*n^2 + 256*n - 1056)*x^2 - 1024*x^3 - 256*x^4)*D[P[n - 1, x], x])/(128*n^3 + 248*n + 64*n*x - 128*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] = ((320*n^3 + 960*n^2 + 1372*n + 732 + (256*n^3 + 768*n^2 + 784*n + 272)*x - (448*n + 448)*x^2 - (256*n + 256)*x^3)*v[i - 1] + (208*n^2 + 208*n + 91 + (640*n^2 + 640*n - 32)*x + (256*n^2 + 256*n - 1056)*x^2 - 1024*x^3 - 256*x^4)*deriv(v[i - 1], x))/(128*n^3 + 248*n + 64*n*x - 128*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 A013611","D+1"],"edges":["e-137d124e39cf8ed1","e-5a19f0d08b31dcc8"],"text":"A116414 \u2192 basis A013611 \u2192 D+1"},"headline":"A116414 \u2192 basis A013611 \u2192 D+1","chain":null}