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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":"12fd73cd7ccee26357645334fd6ba04fb07673f962c35fdeb11278a3517c0087","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-cca535f6115f5fc0","role":"sequence","name":"basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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:      0;\n  1:      2;\n  2:     14,       6;\n  3:     80,      74,      12;\n  4:    416,     592,     228,      20;\n  5:   2048,    3936,    2448,     540,      30;\n  6:   9728,   23552,   20352,    7504,    1090,     42;\n  7:  45056,  131584,  145920,   76416,   19040,   1974,    56;\n  8: 204800,  700416,  949248,  647680,  233280,  42336,  3304,   72;\n  9: 917504, 3596288, 5763072, 4858880, 2309120, 614592, 85344, 5208, 90;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^3-19*n^2+19*n-6-(2*n^3-3*n^2+n)*x)*P(n,x) = -(24*n^3-52*n^2+16*n+12+(10*n^3-39*n^2+17*n+12)*x-(6*n^3-7*n^2-n)*x^2)*P(n-1,x) + ((12*n^2-8*n-4)*x+(8*n^2-8*n-4)*x^2-4*n^2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(24*n^3 - 52*n^2 + 16*n + 12 + (10*n^3 - 39*n^2 + 17*n + 12)*x - (6*n^3 - 7*n^2 - n)*x^2)*P[n - 1, x] + ((12*n^2 - 8*n - 4)*x + (8*n^2 - 8*n - 4)*x^2 - 4*n^2*x^3)*D[P[n - 1, x], x])/(-6*n^3 + 19*n^2 - 19*n + 6 + (2*n^3 - 3*n^2 + n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n^3 - 52*n^2 + 16*n + 12 + (10*n^3 - 39*n^2 + 17*n + 12)*x - (6*n^3 - 7*n^2 - n)*x^2)*v[i - 1] + ((12*n^2 - 8*n - 4)*x + (8*n^2 - 8*n - 4)*x^2 - 4*n^2*x^3)*deriv(v[i - 1], x))/(-6*n^3 + 19*n^2 - 19*n + 6 + (2*n^3 - 3*n^2 + n)*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-c85abc56078671cc","steps":["basis A007318","D"],"edges":["e-2b8d1605acbc4584","e-a313cf5171768a1e"],"text":"A232535 \u2192 basis A007318 \u2192 D"},"headline":"A232535 \u2192 basis A007318 \u2192 D","chain":null}