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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":"7e65f663bb930adad91689b5ab4ef6cbba1a2b42476e4777321a85016d7864bd","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-7fdf5321df1ab451","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator 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31387130640,    145785449160,     205418786400,      64126339200;\n   9:  262092,  253641960,  26553910860,   698551704720,   6099022863360,   19122957833904,   20103889024800,    5056819891200;\n  10: 1048518, 1849403664, 315760216680, 13436013542400, 197127983736360, 1131813079627536, 2584056835942848, 2125321463174400, 443735945452800;\n","recurrence_sentence":"The row polynomials satisfy (3*n-8)*P(n,x) = -(12*n-20+(108*n^2-522*n+570)*x+(972*n^3-7290*n^2+17658*n-13680)*x^2)*P(n-2,x) - ((9*n-15)*x+(162*n^2-729*n+765)*x^2-(486*n^2-2106*n+2160)*x^3)*d/dx P(n-2,x) + (15*n-37+(108*n^2-522*n+624)*x)*P(n-1,x) + ((9*n-24)*x-(27*n-72)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 6.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 20 + (108*n^2 - 522*n + 570)*x + (972*n^3 - 7290*n^2 + 17658*n - 13680)*x^2)*P[n - 2, x] - ((9*n - 15)*x + (162*n^2 - 729*n + 765)*x^2 - (486*n^2 - 2106*n + 2160)*x^3)*D[P[n - 2, x], x] + (15*n - 37 + (108*n^2 - 522*n + 624)*x)*P[n - 1, x] + ((9*n - 24)*x - (27*n - 72)*x^2)*D[P[n - 1, x], x])/(3*n - 8)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 6; for(n = 3, N, my(i = n); v[i] = (-(12*n - 20 + (108*n^2 - 522*n + 570)*x + (972*n^3 - 7290*n^2 + 17658*n - 13680)*x^2)*v[i - 2] - ((9*n - 15)*x + (162*n^2 - 729*n + 765)*x^2 - (486*n^2 - 2106*n + 2160)*x^3)*deriv(v[i - 2], x) + (15*n - 37 + (108*n^2 - 522*n + 624)*x)*v[i - 1] + ((9*n - 24)*x - (27*n - 72)*x^2)*deriv(v[i - 1], x))/(3*n - 8)); 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-7fdf5321df1ab451","steps":["D"],"edges":["e-891150521d3a268d"],"text":"A290316 \u2192 D"},"headline":"A290316 \u2192 D","chain":null}