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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":"b2d3fda588013fe1b043fc861c6291a08db59915aec16dd6fc7987b7b45e2e3e","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-0cecde7ff045a546","role":"sequence","name":"Triangle formed by multiplying Stirling numbers of 2nd kind S2(n,m) (A008277) by m+1.","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:    2;\n  1:    5,     3;\n  2:   11,    17,      4;\n  3:   23,    69,     39,       5;\n  4:   47,   245,    250,      74,       6;\n  5:   95,   813,   1335,     685,     125,      7;\n  6:  191,  2597,   6454,    5110,    1575,    195,     8;\n  7:  383,  8109,  29379,   33705,   15610,   3206,   287,     9;\n  8:  767, 24965, 128650,  205674,  134316,  40698,  5964,   404,  10;\n  9: 1535, 76173, 548895, 1191125, 1054095, 442029, 94290, 10350, 549, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2-x-x^2)*P(n-4,x) - (x-x^2)*d/dx P(n-4,x) - (1+3*x-x^3)*P(n-3,x) - (x+x^2-x^3)*d/dx P(n-3,x) + (3-x-3*x^2)*P(n-2,x) + (x-2*x^2)*d/dx P(n-2,x) + (1+3*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 5 + 3*x, P(2,x) = 11 + 17*x + 4*x^2, P(3,x) = 23 + 69*x + 39*x^2 + 5*x^3.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 5 + 3*x;\nP[2, x_] := 11 + 17*x + 4*x^2;\nP[3, x_] := 23 + 69*x + 39*x^2 + 5*x^3;\nP[n_, x_] := P[n, x] = Expand[-(2 - x - x^2)*P[n - 4, x] - (x - x^2)*D[P[n - 4, x], x] - (1 + 3*x - x^3)*P[n - 3, x] - (x + x^2 - x^3)*D[P[n - 3, x], x] + (3 - x - 3*x^2)*P[n - 2, x] + (x - 2*x^2)*D[P[n - 2, x], x] + (1 + 3*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 3*x; v[3] = 11 + 17*x + 4*x^2; v[4] = 23 + 69*x + 39*x^2 + 5*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(2 - x - x^2)*v[i - 4] - (x - x^2)*deriv(v[i - 4], x) - (1 + 3*x - x^3)*v[i - 3] - (x + x^2 - x^3)*deriv(v[i - 3], x) + (3 - x - 3*x^2)*v[i - 2] + (x - 2*x^2)*deriv(v[i - 2], x) + (1 + 3*x)*v[i - 1] + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["\u03b8+2 (shift +1)","D+1"],"edges":["e-4ce9900b2631a004","e-34950064e1dbba0d"],"text":"A008277 \u2192 \u03b8+2 (shift +1) \u2192 D+1"},"headline":"A008277 \u2192 \u03b8+2 (shift +1) \u2192 D+1","chain":null}