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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":"d444476828df5b6909f0d9acf7529857ddd793df00c2c6a579a0522cbdefbd21","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-a5324901f71bbafe","role":"sequence","name":"basis_map_A008277 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:         10,          2;\n  3:        118,         42,          3;\n  4:       1696,        824,        108,         4;\n  5:      29080,      17400,       3180,       220,        5;\n  6:     581680,     406160,      92880,      9080,      390,       6;\n  7:   13321840,   10505040,    2838360,    350840,    21490,     630,     7;\n  8:  344068480,  299774720,   92393280,  13608000,  1061760,   44688,   952,    8;\n  9: 9898456960, 9381603840, 3221460480, 545039040, 50986320, 2748816, 84504, 1368, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-15*n+6+(6*n-6)*x+x^2)*P(n-2,x) - ((3*n-3)*x+x^2)*d/dx P(n-2,x) + (6*n-2+2*x)*P(n-1,x) + x*d/dx 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[-(9*n^2 - 15*n + 6 + (6*n - 6)*x + x^2)*P[n - 2, x] - ((3*n - 3)*x + x^2)*D[P[n - 2, x], x] + (6*n - 2 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\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] = -(9*n^2 - 15*n + 6 + (6*n - 6)*x + x^2)*v[i - 2] - ((3*n - 3)*x + x^2)*deriv(v[i - 2], x) + (6*n - 2 + 2*x)*v[i - 1] + x*deriv(v[i - 1], 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-8229dbdd6a4b5cd9","steps":["basis A008277","D"],"edges":["e-30984bb1804f505f","e-fa9e3ba8993ad3a5"],"text":"A225470 \u2192 basis A008277 \u2192 D"},"headline":"A225470 \u2192 basis A008277 \u2192 D","chain":null}