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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":"8f27393d1e484ee2bceaad75c83c88eeb7b43dbbd9e6912ec64a706b3a6b896a","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-fdb186db0412847e","role":"sequence","name":"basis_map_A013610 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:          0;\n   2:          6;\n   3:         48,          72;\n   4:        426,        1080,         648;\n   5:       4464,       14832,       15552,        5184;\n   6:      54774,      216720,      304560,      181440,       38880;\n   7:     774432,     3474648,     5909760,     4795200,     1866240,      279936;\n   8:   12427290,    61412904,   119918232,   119206080,    63957600,    17635968,    1959552;\n   9:  223426080,  1193146848,  2591326080,  2985849216,  1981324800,   760306176,  156764160,   13436928;\n  10: 4451852070, 25350903072, 59973517152, 77461743744, 60039040320, 28711355904, 8300662272, 1330255872, 90699264;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(24*n-36)*x+36*x^2)*P(n-2,x) + (4*n-4+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 6.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (24*n - 36)*x + 36*x^2)*P[n - 2, x] + (4*n - 4 + 12*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 6; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (24*n - 36)*x + 36*x^2)*v[i - 2] + (4*n - 4 + 12*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 A013610","D"],"edges":["e-1b98e7733a40a4e4","e-55351b705f21b0c8"],"text":"A161198 \u2192 basis A013610 \u2192 D"},"headline":"A161198 \u2192 basis A013610 \u2192 D","chain":null}