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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":"d43253aecd04a914acdc7a401311481cbc194bb33b477d7d39bdf9eeb8210e80","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-67a3afb389f8b05c","role":"sequence","name":"basis_map_A013609 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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:          1;\n   2:          7,           4;\n   3:         47,          64,          16;\n   4:        389,         764,         432,          64;\n   5:       3921,        9568,        7904,        2560,         256;\n   6:      46911,      132836,      138400,       65920,       14080,       1024;\n   7:     651423,     2060576,     2523184,     1530880,      482560,      73728,      4096;\n   8:   10311885,    35579484,    49178640,    35390656,    14309120,    3232768,    372736,     16384;\n   9:  183410145,   679238208,  1032944064,   845730816,   408364544,  118931456,  20299776,   1835008,   65536;\n  10: 3622630455, 14234969412, 23403465024, 21224395520, 11731646976, 4101888000, 905920512, 121307136, 8847360, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(16*n-24)*x+16*x^2)*P(n-2,x) + (4*n-4+8*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 7 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (16*n - 24)*x + 16*x^2)*P[n - 2, x] + (4*n - 4 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 7 + 4*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (16*n - 24)*x + 16*x^2)*v[i - 2] + (4*n - 4 + 8*x)*v[i - 1]); 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-7c9c430277778c10","steps":["basis A013609","D+1"],"edges":["e-5c9ad73da84eb2a7","e-81c658bd796bd045"],"text":"A161198 \u2192 basis A013609 \u2192 D+1"},"headline":"A161198 \u2192 basis A013609 \u2192 D+1","chain":null}