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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":"30ca03145c60116f157b133b9ab6c072c89b3b554c6fabcdf8876398dfe028cc","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-fc18ad0ea0731402","role":"sequence","name":"Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).","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:           21,             2;\n  1:          343,            84,              3;\n  2:         5145,          2450,            210,             4;\n  3:        74431,         61740,           9555,           420,            5;\n  4:      1058841,       1445402,         360150,         27440,          735,           6;\n  5:     14941423,      32471124,       12252303,       1440600,        65170,        1176,          7;\n  6:    210003465,     711776450,      391771170,      66757404,  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84*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(98 + 14*x + x^2)*v[i - 2] - (49*x + 7*x^2)*deriv(v[i - 2], x) + (21 + 2*x)*v[i - 1] + 7*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-fc18ad0ea0731402","steps":["D"],"edges":["e-0e6f4af46172fa19"],"text":"A075502 \u2192 D"},"headline":"A075502 \u2192 D","chain":null}