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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":"c9e9fae37aef3ab31b60c28df9d432dbc67e55b2253ba67a563f142ff3e50954","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-56eee146d6109546","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 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:        0;\n  1:        1;\n  2:       10,           2;\n  3:       83,          82,            3;\n  4:      668,        2220,          366,            4;\n  5:     5349,       51964,        24630,         1236,           5;\n  6:    42798,     1144734,      1296984,       178860,        3570,          6;\n  7:   342391,    24552990,     60464865,     18543148,     1013445,       9366,        7;\n  8:  2739136,   520406264,   2639571510,   1608403720,   190006460,    4894080,    23086,     8;\n  9: 21913097, 10972357720, 111047126172, 125662813880, 28293143410, 1562943648, 21192416, 54568, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(8*n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (13*n*x+(10*n^2-11*n)*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) - (3*n*x^2+(3*n^2-3*n)*x^3)*d^2/dx^2 P(n-2,x) + (9*n-8+(2*n^2-4*n+2)*x)*P(n-1,x) + ((13*n-13)*x-(n-1)*x^2)*d/dx P(n-1,x) + (3*n-3)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (13*n*x + (10*n^2 - 11*n)*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] - (3*n*x^2 + (3*n^2 - 3*n)*x^3)*D[P[n - 2, x], {x, 2}] + (9*n - 8 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((13*n - 13)*x - (n - 1)*x^2)*D[P[n - 1, x], x] + (3*n - 3)*x^2*D[P[n - 1, x], {x, 2}])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (13*n*x + (10*n^2 - 11*n)*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) - (3*n*x^2 + (3*n^2 - 3*n)*x^3)*deriv(deriv(v[i - 2], x), x) + (9*n - 8 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((13*n - 13)*x - (n - 1)*x^2)*deriv(v[i - 1], x) + (3*n - 3)*x^2*deriv(deriv(v[i - 1], x), x))/(n - 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}}],"derivation":{"root":"n-56eee146d6109546","steps":["D"],"edges":["e-00eeb563f4368218"],"text":"A166972 \u2192 D"},"headline":"A166972 \u2192 D","chain":null}