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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":"1b519f22d05e5f3652b59e74e5756d8508b7ccc5c4c53292a0c9429aa453c150","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-e88a150cc5b6bb0f","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of partitions of an n-set having k blocks of size > 1 (0<=k<=floor(n/2)).","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:   0;\n  2:   1;\n  3:   4;\n  4:  11,     6;\n  5:  26,    50;\n  6:  57,   260,    45;\n  7: 120,  1092,   630;\n  8: 247,  4074,  5250,  420;\n  9: 502, 14142, 34104, 8820;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n^3-6*n^2+11*n-6)*x^2*P(n-4,x) - (2*n^2-6*n+4)*x*P(n-3,x) - (n^2-3*n+2)*x^2*d/dx P(n-3,x) - (2*n-2-(2*n^2-6*n+4)*x)*P(n-2,x) - (n-1)*x*d/dx P(n-2,x) + (3*n-5)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1, P(3,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 6*n^2 + 11*n - 6)*x^2*P[n - 4, x] - (2*n^2 - 6*n + 4)*x*P[n - 3, x] - (n^2 - 3*n + 2)*x^2*D[P[n - 3, x], x] - (2*n - 2 - (2*n^2 - 6*n + 4)*x)*P[n - 2, x] - (n - 1)*x*D[P[n - 2, x], x] + (3*n - 5)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; v[4] = 4; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 - 6*n^2 + 11*n - 6)*x^2*v[i - 4] - (2*n^2 - 6*n + 4)*x*v[i - 3] - (n^2 - 3*n + 2)*x^2*deriv(v[i - 3], x) - (2*n - 2 - (2*n^2 - 6*n + 4)*x)*v[i - 2] - (n - 1)*x*deriv(v[i - 2], x) + (3*n - 5)*v[i - 1] + (n - 2)*x*deriv(v[i - 1], x))/(n - 2)); 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-e88a150cc5b6bb0f","steps":["D"],"edges":["e-67de9e2032b24ea2"],"text":"A124324 \u2192 D"},"headline":"A124324 \u2192 D","chain":null}