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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":"518ed162b2e99b9be32fad65c83194b972b78b246ccda5114dacbdca082f7995","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-317113238d23620b","role":"sequence","name":"basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:       1;\n  1:       6,       1;\n  2:      32,      12,       1;\n  3:     164,      99,      18,       1;\n  4:     819,     692,     201,      24,      1;\n  5:    4012,    4392,    1784,     338,     30,      1;\n  6:   19363,   26148,   13779,    3640,    510,     36,     1;\n  7:   92352,  148717,   96870,   33165,   6460,    717,    42,    1;\n  8:  436256,  817260,  636205,  269240,  67860,  10444,   959,   48,  1;\n  9: 2044430, 4372455, 3968280, 2008110, 625170, 124299, 15792, 1236, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+2*n-14*n*x-2*n*x^2)*P(n,x) = (5*n^3+15*n^2+7*n-3+(n^3+3*n^2-41*n-43)*x-(13*n+13)*x^2-(n+1)*x^3)*P(n-1,x) + (13*n^2+13*n-104+(10*n^2+10*n-197)*x+(n^2+n-111)*x^2-19*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n^3 + 15*n^2 + 7*n - 3 + (n^3 + 3*n^2 - 41*n - 43)*x - (13*n + 13)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (13*n^2 + 13*n - 104 + (10*n^2 + 10*n - 197)*x + (n^2 + n - 111)*x^2 - 19*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 + 2*n - 14*n*x - 2*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n^3 + 15*n^2 + 7*n - 3 + (n^3 + 3*n^2 - 41*n - 43)*x - (13*n + 13)*x^2 - (n + 1)*x^3)*v[i - 1] + (13*n^2 + 13*n - 104 + (10*n^2 + 10*n - 197)*x + (n^2 + n - 111)*x^2 - 19*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 + 2*n - 14*n*x - 2*n*x^2)); 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-6466a8550c08a52f","steps":["basis A007318","D+1"],"edges":["e-235ea1c8ecbb7de6","e-9badce58e627b9a0"],"text":"A116414 \u2192 basis A007318 \u2192 D+1"},"headline":"A116414 \u2192 basis A007318 \u2192 D+1","chain":null}