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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":"73264d7b0ba60bde5001c9d5db032174aa3b21e49d80a1590d1d8f28de5fcd30","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-efe23a76953cf69d","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. 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v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (2*n - 2)*x + x^2)*v[i - 2] + (2*n - 1 + 2*x)*v[i - 1]); 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}},{"oeis_id":"A074246","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        3,        2;\n   3:       11,       12,        3;\n   4:       50,       70,       30,        4;\n   5:      274,      450,      255,       60,       5;\n   6:     1764,     3248,     2205,      700,     105,      6;\n   7:    13068,    26264,    20307,     7840,    1610,    168,      7;\n   8:   109584,   236248,   201852,    89796,   22680,   3276,    252,     8;\n   9:  1026576,  2345400,  2171040,  1077300,  316365,  56700,   6090,   360,   9;\n  10: 10628640, 25507152, 25228500, 13667720, 4510275, 946638, 127050, 10560, 495, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(2*n-2)*x+x^2)*P(n-2,x) + (2*n-1+2*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (2*n - 2)*x + x^2)*P[n - 2, x] + (2*n - 1 + 2*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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