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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":"7b5e7f81de49a9965f7f45ecf39a250b1321de6a415cac5369849891dd507d49","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-adb5eb667a229c4c","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 6).","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:           18,             2;\n  1:          252,            72,             3;\n  2:         3240,          1800,           180,             4;\n  3:        40176,         38880,          7020,           360,            5;\n  4:       489888,        780192,        226800,         20160,          630,           6;\n  5:      5925312,      15023232,       6613488,        907200,        47880,        1008,          7;\n  6:     71383680,     282268800,     181258560,      36033984,      2857680,       99792,       1512,        8;\n  7:    858283776,    5223605760,    4773608640,    1322697600,    147918960,     7620480,     189000,     2160,      9;\n  8:  10309483008,   95741471232,  122402016000,   46045739520,   6978454560,   497562912,   17962560,   332640,   2970,   10;\n  9: 123774262272, 1743965448192, 3081260590848, 1544574873600, 308781538560, 29271787776, 1442692944, 38491200, 552420, 3960, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(72+12*x+x^2)*P(n-2,x) - (36*x+6*x^2)*d/dx P(n-2,x) + (18+2*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 18 + 2*x, P(1,x) = 252 + 72*x + 3*x^2.","mathematica":"P[0, x_] := 18 + 2*x;\nP[1, x_] := 252 + 72*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(72 + 12*x + x^2)*P[n - 2, x] - (36*x + 6*x^2)*D[P[n - 2, x], x] + (18 + 2*x)*P[n - 1, x] + 6*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] = 18 + 2*x; v[2] = 252 + 72*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(72 + 12*x + x^2)*v[i - 2] - (36*x + 6*x^2)*deriv(v[i - 2], x) + (18 + 2*x)*v[i - 1] + 6*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-adb5eb667a229c4c","steps":["D"],"edges":["e-d20098ac54dd6525"],"text":"A075501 \u2192 D"},"headline":"A075501 \u2192 D","chain":null}