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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":"4a48302c701ef7ef3ba8d0940b2f742aa9e8fd4aacef4778c1baddf4a4549900","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-f074ae9ffad82cf4","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","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:    0;\n  2:    6;\n  3:   12;\n  4:   54,    72;\n  5:  132,   216;\n  6:  408,  1080,   648;\n  7: 1044,  3312,  2592;\n  8: 2850, 11448, 14256,  5184;\n  9: 7284, 33912, 51840, 25920;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4)*P(n,x) = (14*n-38+(24*n-72)*x)*P(n-1,x) - (13+50*x+48*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((14*n - 38 + (24*n - 72)*x)*P[n - 1, x] - (13 + 50*x + 48*x^2)*D[P[n - 1, x], x])/(2*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((14*n - 38 + (24*n - 72)*x)*v[i - 1] - (13 + 50*x + 48*x^2)*deriv(v[i - 1], x))/(2*n - 4)); 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-e189038e95d51606","steps":["basis A013609","D"],"edges":["e-79f6722e524950c5","e-76e167a87af6a8d3"],"text":"A304236 \u2192 basis A013609 \u2192 D"},"headline":"A304236 \u2192 basis A013609 \u2192 D","chain":null}