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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":"84821082432a854612a03cb29e8cb23dd9b30bea5706ef3557c0f067ae2ea332","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0892e4c474e4320d","role":"sequence","name":"basis_map_A013610 of Riordan array (1, 2+2x).","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:      8,       6;\n  2:     36,     102,       36;\n  3:    136,     696,      936,       216;\n  4:    476,    3528,     8676,      7560,      1296;\n  5:   1584,   15432,    55584,     88776,     57024,      7776;\n  6:   5104,   61656,   293472,    689688,    811296,    412128,     46656;\n  7:  16064,  231744,  1373184,   4278528,   7418304,   6889536,   2892672,    279936;\n  8:  49680,  833472,  5914080,  22996224,  52955856,  72518976,  55567296,  19875456,   1679616;\n  9: 151552, 2899296, 23974272, 111844800, 321916032, 586007136, 661955328, 431381376, 134369280, 10077696;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-3*n^2-4*n+(6*n^2-22*n)*x+(18*n^2-24*n)*x^2+18*n*x^3)*P(n,x) = -(6*n^3-42*n^2+40*n+28-(42*n^2-58*n-136)*x+(36*n^2-48*n+156)*x^2-(72*n+36)*x^3-108*x^4)*P(n-1,x) + (6*n^2-6*n-4+(24*n^2-12*n-24)*x+(18*n^2+66*n-20)*x^2+(180*n+108)*x^3+(108*n+216)*x^4+108*x^5)*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[(-(6*n^3 - 42*n^2 + 40*n + 28 - (42*n^2 - 58*n - 136)*x + (36*n^2 - 48*n + 156)*x^2 - (72*n + 36)*x^3 - 108*x^4)*P[n - 1, x] + (6*n^2 - 6*n - 4 + (24*n^2 - 12*n - 24)*x + (18*n^2 + 66*n - 20)*x^2 + (180*n + 108)*x^3 + (108*n + 216)*x^4 + 108*x^5)*D[P[n - 1, x], x])/(3*n^3 - 3*n^2 - 4*n + (6*n^2 - 22*n)*x + (18*n^2 - 24*n)*x^2 + 18*n*x^3)]];\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] = (-(6*n^3 - 42*n^2 + 40*n + 28 - (42*n^2 - 58*n - 136)*x + (36*n^2 - 48*n + 156)*x^2 - (72*n + 36)*x^3 - 108*x^4)*v[i - 1] + (6*n^2 - 6*n - 4 + (24*n^2 - 12*n - 24)*x + (18*n^2 + 66*n - 20)*x^2 + (180*n + 108)*x^3 + (108*n + 216)*x^4 + 108*x^5)*deriv(v[i - 1], x))/(3*n^3 - 3*n^2 - 4*n + (6*n^2 - 22*n)*x + (18*n^2 - 24*n)*x^2 + 18*n*x^3)); 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-f5a001953ee28e8f","steps":["basis A013610","D+1"],"edges":["e-937d6a83cea8d959","e-4260492f5c15605b"],"text":"A099040 \u2192 basis A013610 \u2192 D+1"},"headline":"A099040 \u2192 basis A013610 \u2192 D+1","chain":null}