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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":"86ee67a2a1a4b188b7de9047c4d51dc553d0653fd164c32e1be95f6c1e50be3a","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-945ed98594b7abe2","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","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:     0;\n   2:     2;\n   3:    16,       24;\n   4:    66,      568,        360;\n   5:   224,     5744,      17856,        6720;\n   6:   706,    42160,     349008,      584256,       151200;\n   7:  2160,   264456,    4593216,    19218624,     20651520,       3991680;\n   8:  6530,  1521000,   48813240,   405690560,   1043958240,     795346560,     121080960;\n 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11)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(12*n - 21 + (64*n^2 - 320*n + 364)*x + (256*n^3 - 1984*n^2 + 4976*n - 4004)*x^2)*v[i - 2] - ((8*n - 14)*x + (64*n^2 - 304*n + 336)*x^2 - (128*n^2 - 576*n + 616)*x^3)*deriv(v[i - 2], x) + (16*n - 40 + (64*n^2 - 320*n + 396)*x)*v[i - 1] + ((8*n - 22)*x - (16*n - 44)*x^2)*deriv(v[i - 1], x))/(4*n - 11)); 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-945ed98594b7abe2","steps":["D"],"edges":["e-e9f8baaddb18df8b"],"text":"A290315 \u2192 D"},"headline":"A290315 \u2192 D","chain":null}