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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":"fff1acab73ab6583b62dec680557632ebfa1b132d6c96de303805365215df1d0","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-9aa9e3dd2403bca2","role":"sequence","name":"shift_x_plus_one of Riordan array ((1+x)^3,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: 12,  12,    3;\n  1: 20,  36,   21,    4;\n  2: 28,  76,   75,   32,     5;\n  3: 36, 132,  189,  132,    45,     6;\n  4: 44, 204,  387,  384,   210,    60,     7;\n  5: 52, 292,  693,  900,   690,   312,    77,    8;\n  6: 60, 396, 1131, 1824,  1815,  1140,   441,   96,    9;\n  7: 68, 516, 1725, 3332,  4095,  3318,  1771,  600,  117,   10;\n  8: 76, 652, 2499, 5632,  8260,  8232,  5642, 2624,  792,  140,  11;\n  9: 84, 804, 3477, 8964, 15300, 18144, 15246, 9072, 3744, 1020, 165, 12;\n","recurrence_sentence":"The row polynomials satisfy (2*n+1+(n+2)*x)*P(n,x) = (2*n+3+(3*n+6)*x+(n+3)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 12*x + 3*x^2.","mathematica":"P[0, x_] := 12 + 12*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 3 + (3*n + 6)*x + (n + 3)*x^2)*P[n - 1, x])/(2*n + 1 + (n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 12 + 12*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 3 + (3*n + 6)*x + (n + 3)*x^2)*v[i - 1])/(2*n + 1 + (n + 2)*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-f48bed27dbc29c5f","steps":["P\u2099 + P\u2099\u208a\u2081","D"],"edges":["e-0331d381cbbd8a41","e-9d6c253abd017654"],"text":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081 \u2192 D"},"headline":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081 \u2192 D","chain":null}