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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":"dced20633c89eb751e3ba8ed8537ba78a8c7fb854485cdbeb7dd033ffaf0cc63","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-e734a631f8279507","role":"sequence","name":"Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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:   1;\n  2:   5,     8;\n  3:  14,    98,     108;\n  4:  30,   546,    2460,      2304;\n  5:  55,  2046,   22935,     84304,      72000;\n  6:  91,  6006,  133419,   1185184,    3865680,     3110400;\n  7: 140, 14924,  574860,   9901892,   76458200,   230412384,    177811200;\n  8: 204, 32844, 2007564,  58956612,  868609560,  6102402144,  17381935872,   13005619200;\n  9: 285, 65892, 5998110, 275773524, 6837966525, 90531251376, 594058938480, 1622076254208, 1185137049600;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2+(2*n^4-4*n^3+2*n^2)*x+(n^6-4*n^5+6*n^4-4*n^3+n^2)*x^2)*P(n-2,x) + (2*n^2-2*n+1+(2*n^4-4*n^3+2*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + (2*n^4 - 4*n^3 + 2*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + (2*n^4 - 4*n^3 + 2*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*v[i - 2] + (2*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*v[i - 1])/(n^2 - 2*n + 1)); 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-e734a631f8279507","steps":["D"],"edges":["e-d8a0e8109edcc8ad"],"text":"A008955 \u2192 D"},"headline":"A008955 \u2192 D","chain":null}