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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":"1c2b41e8efcb673401a357f322fc5c9d2c45aa1ec2ea74051164b59d8515ff59","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1eae5c7270ad0739","role":"sequence","name":"Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:      -2,       4,        2;\n   2:     -11,     -11,       24,         8;\n   3:       8,    -148,      -74,       192,          48;\n   4:     119,     119,    -1896,      -632,        1920,        384;\n   5:     -48,    3268,     1634,    -26432,       -6608,      23040,        3840;\n   6:   -1409,   -1409,    72096,     24032,     -409920,     -81984,      322560,       46080;\n   7:     384,  -65596,   -32798,   1557248,      389312,   -7070976,    -1178496,     5160960,       645120;\n   8:   18825,   18825, -2223168,   -741056,    34834240,    6966848,  -134894592,   -19270656,     92897280,    10321920;\n   9:   -3840, 1333660,   666830, -68928512,   -17232128,  823129344,   137188224, -2826584064,   -353323008,  1857945600,   185794560;\n  10: -283875, -283875, 64464648,  21488216, -2106389440, -421277888, 20690191872,  2955741696, -64620011520, -7180001280, 40874803200, 3715891200;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n^3-n^2)*P(n-4,x) + (4*n^3-8*n^2)*x*P(n-3,x) - (2*n^2-n-2+(4*n^3-12*n^2+8*n)*x^2)*P(n-2,x) + (4*n^2-8*n)*x*P(n-1,x) for n >= 5, with P(1,x) = -2 + 4*x + 2*x^2, P(2,x) = -11 - 11*x + 24*x^2 + 8*x^3, P(3,x) = 8 - 148*x - 74*x^2 + 192*x^3 + 48*x^4, P(4,x) = 119 + 119*x - 1896*x^2 - 632*x^3 + 1920*x^4 + 384*x^5.","mathematica":"P[1, x_] := -2 + 4*x + 2*x^2;\nP[2, x_] := -11 - 11*x + 24*x^2 + 8*x^3;\nP[3, x_] := 8 - 148*x - 74*x^2 + 192*x^3 + 48*x^4;\nP[4, x_] := 119 + 119*x - 1896*x^2 - 632*x^3 + 1920*x^4 + 384*x^5;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2)*P[n - 4, x] + (4*n^3 - 8*n^2)*x*P[n - 3, x] - (2*n^2 - n - 2 + (4*n^3 - 12*n^2 + 8*n)*x^2)*P[n - 2, x] + (4*n^2 - 8*n)*x*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = -2 + 4*x + 2*x^2; v[2] = -11 - 11*x + 24*x^2 + 8*x^3; v[3] = 8 - 148*x - 74*x^2 + 192*x^3 + 48*x^4; v[4] = 119 + 119*x - 1896*x^2 - 632*x^3 + 1920*x^4 + 384*x^5; for(n = 5, N, my(i = n); v[i] = (-(n^3 - n^2)*v[i - 4] + (4*n^3 - 8*n^2)*x*v[i - 3] - (2*n^2 - n - 2 + (4*n^3 - 12*n^2 + 8*n)*x^2)*v[i - 2] + (4*n^2 - 8*n)*x*v[i - 1])/(n - 2)); 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-1eae5c7270ad0739","steps":["D+1"],"edges":["e-e2c70df937a9aaeb"],"text":"A136668 \u2192 D+1"},"headline":"A136668 \u2192 D+1","chain":null}