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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":"04f7c4e196fe04e83aa0a278ee5d14f93d0a472ff85a54e2d82911372e7965e5","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-b17a463cc18bc314","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 4).","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:         12,           2;\n  1:        112,          48,            3;\n  2:        960,         800,          120,           4;\n  3:       7936,       11520,         3120,         240,           5;\n  4:      64512,      154112,        67200,        8960,         420,          6;\n  5:     520192,     1978368,      1306368,      268800,       21280,        672,         7;\n  6:    4177920,    24780800,     23869440,     7117824,      846720,      44352,      1008,        8;\n  7:   33488896,   305725440,    419082240,   174182400,    29218560,    2257920,     84000,     1440,      9;\n  8:  268173312,  3735683072,   7163904000,  4042424320,   918973440,   98284032,   5322240,   147840,   1980,   10;\n  9: 2146435072, 45364543488, 120225988608, 90400358400, 27108392960, 3854721024, 284976384, 11404800, 245520, 2640, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(32+8*x+x^2)*P(n-2,x) - (16*x+4*x^2)*d/dx P(n-2,x) + (12+2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 12 + 2*x, P(1,x) = 112 + 48*x + 3*x^2.","mathematica":"P[0, x_] := 12 + 2*x;\nP[1, x_] := 112 + 48*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(32 + 8*x + x^2)*P[n - 2, x] - (16*x + 4*x^2)*D[P[n - 2, x], x] + (12 + 2*x)*P[n - 1, x] + 4*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] = 12 + 2*x; v[2] = 112 + 48*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(32 + 8*x + x^2)*v[i - 2] - (16*x + 4*x^2)*deriv(v[i - 2], x) + (12 + 2*x)*v[i - 1] + 4*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-b17a463cc18bc314","steps":["D"],"edges":["e-b4299bd07c9e8699"],"text":"A075499 \u2192 D"},"headline":"A075499 \u2192 D","chain":null}