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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"d9c1348cd7198546b758a6e40c70984f1325763a830b5f2e1354c5f3a8c56eb4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b58ba5723c3a3a3f","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","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:       24,         18;\n   2:      141,        282,          81;\n   3:      726,       2820,        2178,          324;\n   4:     3471,      22812,       34218,        13884,         1215;\n   5:    15828,     162654,      410616,       325308,        79140,         4374;\n   6:    69873,    1066518,     4171563,      5562084,      2666295,       419238,        15309;\n   7:   301362,    6590304,    37827594,     77869176,     63045990,     19770912,      2109534,       52488;\n   8:  1277619,   38978760,   315934668,    948578568,   1185723210,    631869336,    136425660,    10220952,     177147;\n   9:  5346672,  223001418,  2480289408,  10431265992,  18971571360,  15646898988,   5787341952,   892005672,   48120048,    590490;\n  10: 22154325, 1243327218, 18561259845, 106089387384, 269124313290, 322949175948, 185656427922, 49496692920, 5594972481, 221543250, 1948617;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(12*n+36+(6*n^2+24*n+18)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((3*n+9)*x+(n^2+2*n-3)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (7*n+17+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 24 + 18*x, P(2,x) = 141 + 282*x + 81*x^2.","mathematica":"P[1, x_] := 24 + 18*x;\nP[2, x_] := 141 + 282*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 36 + (6*n^2 + 24*n + 18)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((3*n + 9)*x + (n^2 + 2*n - 3)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (7*n + 17 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 24 + 18*x; v[2] = 141 + 282*x + 81*x^2; for(n = 3, N, my(i = n); v[i] = (-(12*n + 36 + (6*n^2 + 24*n + 18)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*v[i - 2] - ((3*n + 9)*x + (n^2 + 2*n - 3)*x^2 - (n^2 + 5*n + 6)*x^3)*deriv(v[i - 2], x) + (7*n + 17 + (2*n^2 + 8*n + 8)*x)*v[i - 1] + ((n + 2)*x - (n + 2)*x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b58ba5723c3a3a3f","steps":["D"],"edges":["e-282fad3be832133b"],"text":"A257180 \u2192 D"},"headline":"A257180 \u2192 D","chain":null}