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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":"2c839f1c3ebc3e1d828c90031096440f6e1e24edf5f581aceb3fc9b9ba35961a","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-d8e527df92e2a449","role":"sequence","name":"basis_map_A013610 of Riordan array (1+2x,x(1+x)).","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:   15,    18;\n  3:   39,   126,     81;\n  4:   87,   468,    729,    324;\n  5:  180,  1350,   3483,   3564,    1215;\n  6:  354,  3420,  12393,  20736,   15795,    4374;\n  7:  675,  7974,  37341,  87804,  108135,   65610,   15309;\n  8: 1257, 17568, 101007, 307476,  530955,  516132,  260253,  52488;\n  9: 2301, 37134, 253287, 948672, 2121390, 2882466, 2311659, 997272, 177147;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-12*n^2+9*n-(9*n^2-9*n)*x)*P(n,x) = -(6*n^3-57*n^2+135*n-60-(45*n^2-171*n+36)*x+54*n*x^2)*P(n-1,x) + (5*n^2-20*n+10+(18*n^2-87*n+36)*x+(9*n^2-90*n+18)*x^2-27*n*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[(-(6*n^3 - 57*n^2 + 135*n - 60 - (45*n^2 - 171*n + 36)*x + 54*n*x^2)*P[n - 1, x] + (5*n^2 - 20*n + 10 + (18*n^2 - 87*n + 36)*x + (9*n^2 - 90*n + 18)*x^2 - 27*n*x^3)*D[P[n - 1, x], x])/(3*n^3 - 12*n^2 + 9*n - (9*n^2 - 9*n)*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] = (-(6*n^3 - 57*n^2 + 135*n - 60 - (45*n^2 - 171*n + 36)*x + 54*n*x^2)*v[i - 1] + (5*n^2 - 20*n + 10 + (18*n^2 - 87*n + 36)*x + (9*n^2 - 90*n + 18)*x^2 - 27*n*x^3)*deriv(v[i - 1], x))/(3*n^3 - 12*n^2 + 9*n - (9*n^2 - 9*n)*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-c27b11147fce8bfa","steps":["basis A013610","D"],"edges":["e-849840c29c946dc7","e-a4cb06251d15d9d4"],"text":"A113214 \u2192 basis A013610 \u2192 D"},"headline":"A113214 \u2192 basis A013610 \u2192 D","chain":null}