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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":"ac0689739bff5ef0b58fabafbcdc77f00355388d53e6a892475614bb67560d85","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-ee73bfb592710e97","role":"sequence","name":"Triangle read by rows, (2, 2)-Lah numbers.","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:               52,                2;\n  3:             2948,              232,               3;\n  4:           203072,            24688,             648,             4;\n  5:         17154432,          2874944,          115440,          1440,            5;\n  6:       1760601600,        382177088,        20935776,        397280,         2780,          6;\n  7:     216690624000,      58554703872,      4111411776,     105781248,      1122100,       4872,       7;\n  8:   31569232896000,   10329126202368,    893668763904,   29282663168,    419484160,    2754528,    7952,     8;\n  9: 5379460107264000, 2087647201456128, 216569161181952, 8687920122880, 157834251200, 1395848064, 6084288, 12288, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4+10*n^3+37*n^2+60*n+36+(2*n^2+8*n+8)*x+x^2)*P(n-2,x) - ((2*n^3+15*n^2+36*n+28)*x+(2*n+6)*x^2)*d/dx P(n-2,x) - ((n^2+4*n+4)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2+12*n+20+2*x)*P(n-1,x) + (2*n+9)*x*d/dx P(n-1,x) + 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[-(n^4 + 10*n^3 + 37*n^2 + 60*n + 36 + (2*n^2 + 8*n + 8)*x + x^2)*P[n - 2, x] - ((2*n^3 + 15*n^2 + 36*n + 28)*x + (2*n + 6)*x^2)*D[P[n - 2, x], x] - ((n^2 + 4*n + 4)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 12*n + 20 + 2*x)*P[n - 1, x] + (2*n + 9)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\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] = -(n^4 + 10*n^3 + 37*n^2 + 60*n + 36 + (2*n^2 + 8*n + 8)*x + x^2)*v[i - 2] - ((2*n^3 + 15*n^2 + 36*n + 28)*x + (2*n + 6)*x^2)*deriv(v[i - 2], x) - ((n^2 + 4*n + 4)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 12*n + 20 + 2*x)*v[i - 1] + (2*n + 9)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-ee73bfb592710e97","steps":["D"],"edges":["e-4a5e833d8061efc3"],"text":"A371081 \u2192 D"},"headline":"A371081 \u2192 D","chain":null}