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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":"552698eff4bdc8fe6808608a1b5b2b93fea2b4a6b9d47f67942eecb0c35a26c9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2c8ebdd6fbbf3af3","role":"sequence","name":"shift_x_plus_one of Triangle of B-analogs of Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,        1;\n   3:      12,        8,        1;\n   4:      58,       58,       15,        1;\n   5:     324,      432,      172,       24,       1;\n   6:    2044,     3396,     1840,      400,      35,      1;\n   7:   14320,    28384,    19516,     5840,     800,     48,     1;\n   8:  109960,   252456,   210476,    80836,   15330,   1442,    63,    1;\n   9:  916112,  2385280,  2335712,  1099840,  269416,  35168,  2408,   80,  1;\n  10: 8215088, 23874448, 26825472, 14992384, 4533816, 766920, 72912, 3792, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2-2*x-x^2)*P(n-2,x) + (2*x+2*x^2)*d/dx P(n-2,x) + 4*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-(2 - 2*x - x^2)*P[n - 2, x] + (2*x + 2*x^2)*D[P[n - 2, x], x] + 4*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(2 - 2*x - x^2)*v[i - 2] + (2*x + 2*x^2)*deriv(v[i - 2], x) + 4*v[i - 1] + (2 + 2*x)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d55b2cfe05a04774","steps":["x \u21a6 x+1","D+1"],"edges":["e-25255a5163ba8ab9","e-8e18ddc1d9b2321a"],"text":"A039755 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A039755 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}