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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":"88df0d67874120f54b19838eb6e5ed2aad02aba684f5e6754c66edf90b4c72fc","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-ad5020a0c2e8a31d","role":"sequence","name":"basis_map_A013610 of A Jacobsthal triangle.","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:    3;\n  2:   12,    18;\n  3:   33,   126,     81;\n  4:   90,   504,    810,     324;\n  5:  219,  1692,   4374,    4212,    1215;\n  6:  528,  5094,  18144,   28836,   19440,    4374;\n  7: 1221, 14274,  65043,  144828,  161595,   83106,   15309;\n  8: 2790, 38124, 211410,  607824,  959850,  813564,  336798,   52488;\n  9: 6255, 98064, 641520, 2263464, 4648590, 5598720, 3796632, 1312200, 177147;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n+1-(3*n-3)*x)*P(n,x) = -(8*n^2-42*n+40-(30*n-54)*x+18*x^2)*P(n-1,x) + (6*n-9+(20*n-39)*x+(6*n-39)*x^2-9*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 42*n + 40 - (30*n - 54)*x + 18*x^2)*P[n - 1, x] + (6*n - 9 + (20*n - 39)*x + (6*n - 39)*x^2 - 9*x^3)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1 - (3*n - 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^2 - 42*n + 40 - (30*n - 54)*x + 18*x^2)*v[i - 1] + (6*n - 9 + (20*n - 39)*x + (6*n - 39)*x^2 - 9*x^3)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1 - (3*n - 3)*x)); 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-ba196dcd01800741","steps":["basis A013610","D"],"edges":["e-900137e7780f9cba","e-eccaae617e842576"],"text":"A113953 \u2192 basis A013610 \u2192 D"},"headline":"A113953 \u2192 basis A013610 \u2192 D","chain":null}