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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":"44bf7325f88e58ba445e133a8fae86b394f88ead554cb96b08136a860dda8d0c","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-0026e6b972fa04d4","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 9).","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:            27,              2;\n  1:           567,            108,              3;\n  2:         10935,           4050,            270,              4;\n  3:        203391,         131220,          15795,            540,             5;\n  4:       3720087,        3949722,         765450,          45360,           945,            6;\n  5:      67493007,      114082668,       33480783,        3061800,        107730,         1512,          7;\n  6:    1219657095,     3215218050,     1376432190,      182422044,       9644670,       224532,       2268,         8;\n  7:   21996874431,    89250201540,    54374385915,    10044234900,     748839735,     25719120,     425250,      3240,       9;\n  8:  396331160247,  2453749190442,  2091353195250,   524489751720,   52992639315,   2518912242,   60623640,    748440,    4455,   10;\n  9: 7137447668847, 67043890462428, 78969338814663, 26390509754400, 3517214712660, 222282638424, 7303633029, 129907800, 1242945, 5940, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(162+18*x+x^2)*P(n-2,x) - (81*x+9*x^2)*d/dx P(n-2,x) + (27+2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 27 + 2*x, P(1,x) = 567 + 108*x + 3*x^2.","mathematica":"P[0, x_] := 27 + 2*x;\nP[1, x_] := 567 + 108*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(162 + 18*x + x^2)*P[n - 2, x] - (81*x + 9*x^2)*D[P[n - 2, x], x] + (27 + 2*x)*P[n - 1, x] + 9*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] = 27 + 2*x; v[2] = 567 + 108*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(162 + 18*x + x^2)*v[i - 2] - (81*x + 9*x^2)*deriv(v[i - 2], x) + (27 + 2*x)*v[i - 1] + 9*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-0026e6b972fa04d4","steps":["D"],"edges":["e-57544a8dd9fb3ec1"],"text":"A075504 \u2192 D"},"headline":"A075504 \u2192 D","chain":null}