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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":"b3341e55222bcbd02e50a3cc258a4a39f6dbfd2841100b81428ef7bae70400fb","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-1e0cd3ca23b1a82f","role":"sequence","name":"basis_map_A013609 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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:        10,         16;\n  3:        68,        200,         144;\n  4:       596,       2488,        3408,        1536;\n  5:      6414,      34312,       67992,       59264,       19200;\n  6:     81894,     531368,     1365528,     1739776,     1100160,      276480;\n  7:   1210152,    9200816,    28910592,    48106944,    44748160,    22076928,     4515840;\n  8:  20314440,  176737104,   654478464,  1338363456,  1633361280,  1190318592,   479840256,    82575360;\n  9: 381831930, 3735850608, 15898035696, 38459661888, 57880319520, 55516019712, 33153974784, 11274780672, 1672151040;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(8*n^3-20*n^2+12*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(4*n^2-4*n)*x)*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[(-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*P[n - 1, x])/(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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*v[i - 1])/(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-849de7332df28657","steps":["basis A013609","D"],"edges":["e-0b105283f866e3e7","e-c6123fcd7f6af0a8"],"text":"A088996 \u2192 basis A013609 \u2192 D"},"headline":"A088996 \u2192 basis A013609 \u2192 D","chain":null}