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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"70f68a8c9455795081433893c09b3a0cbac319900565091791fbd43492057f03","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d0e888e78d074358","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:           0;\n   2:           2;\n   3:          19,            8;\n   4:         222,          294,           24;\n   5:        3194,         8256,         2661,           64;\n   6:       55024,       227132,       157614,        19340,          160;\n   7:     1107336,      6537576,      7688028,      2222736,       125835,         384;\n   8:    25526192,    201088824,    352553880,    181662560,     26505750,      769218,        896;\n   9:   663605680,   6651534752,  16056935292,  12757532800,   3478572950,   285223920,    4525633,     2048;\n  10: 19213911360, 236723438592, 745481842488, 835985913552, 362399742000, 58181791500, 2870269038, 25961112, 4608;\n","recurrence_sentence":"The row polynomials satisfy (3*n-7)*P(n,x) = -(27*n^3-153*n^2+276*n-160+(54*n^3-396*n^2+912*n-640)*x+(27*n^3-189*n^2+414*n-280)*x^2)*P(n-2,x) - ((27*n^2-108*n+96)*x+(9*n-12)*x^2-(27*n^2-99*n+84)*x^3)*d/dx P(n-2,x) + (18*n^2-63*n+46+(18*n^2-84*n+98)*x)*P(n-1,x) + ((9*n-21)*x-(9*n-21)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 153*n^2 + 276*n - 160 + (54*n^3 - 396*n^2 + 912*n - 640)*x + (27*n^3 - 189*n^2 + 414*n - 280)*x^2)*P[n - 2, x] - ((27*n^2 - 108*n + 96)*x + (9*n - 12)*x^2 - (27*n^2 - 99*n + 84)*x^3)*D[P[n - 2, x], x] + (18*n^2 - 63*n + 46 + (18*n^2 - 84*n + 98)*x)*P[n - 1, x] + ((9*n - 21)*x - (9*n - 21)*x^2)*D[P[n - 1, x], x])/(3*n - 7)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(27*n^3 - 153*n^2 + 276*n - 160 + (54*n^3 - 396*n^2 + 912*n - 640)*x + (27*n^3 - 189*n^2 + 414*n - 280)*x^2)*v[i - 2] - ((27*n^2 - 108*n + 96)*x + (9*n - 12)*x^2 - (27*n^2 - 99*n + 84)*x^3)*deriv(v[i - 2], x) + (18*n^2 - 63*n + 46 + (18*n^2 - 84*n + 98)*x)*v[i - 1] + ((9*n - 21)*x - (9*n - 21)*x^2)*deriv(v[i - 1], x))/(3*n - 7)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d0e888e78d074358","steps":["D"],"edges":["e-0ef18a410d75d745"],"text":"A290595 \u2192 D"},"headline":"A290595 \u2192 D","chain":null}