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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":"92c5c1b946dfff3d61a2f12dba53deebc44772445a60ab4d28579df72b242c87","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-e224ad3e3147ef52","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.","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:         4;\n  2:        56,         32;\n  3:       552,       1104,         192;\n  4:      4696,      22080,       14088,         1024;\n  5:     36968,     342896,      514344,       147872,         5120;\n  6:    278232,    4611936,    13372944,      9223872,      1391160,        24576;\n  7:   2037736,   56692176,   284534736,    379379648,    141730440,     12226416,      114688;\n  8:  14673752,  656582144,  5314678584,  12140148736,   8857797640,   1969746432,   102716264,    524288;\n  9: 104551272, 7299509040, 90758032344, 328880903904, 411101129880, 181516064688, 25548281640, 836410176, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (3*n-2)*P(n,x) = -(84*n+28+(72*n^2-96*n-40)*x+(27*n^3-54*n^2+9*n+10)*x^2)*P(n-2,x) - ((36*n+12)*x+(27*n^2-45*n-18)*x^2-(27*n^2-9*n-6)*x^3)*d/dx P(n-2,x) + (33*n-10+(18*n^2-24*n+8)*x)*P(n-1,x) + ((9*n-6)*x-(9*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(84*n + 28 + (72*n^2 - 96*n - 40)*x + (27*n^3 - 54*n^2 + 9*n + 10)*x^2)*P[n - 2, x] - ((36*n + 12)*x + (27*n^2 - 45*n - 18)*x^2 - (27*n^2 - 9*n - 6)*x^3)*D[P[n - 2, x], x] + (33*n - 10 + (18*n^2 - 24*n + 8)*x)*P[n - 1, x] + ((9*n - 6)*x - (9*n - 6)*x^2)*D[P[n - 1, x], x])/(3*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(84*n + 28 + (72*n^2 - 96*n - 40)*x + (27*n^3 - 54*n^2 + 9*n + 10)*x^2)*v[i - 2] - ((36*n + 12)*x + (27*n^2 - 45*n - 18)*x^2 - (27*n^2 - 9*n - 6)*x^3)*deriv(v[i - 2], x) + (33*n - 10 + (18*n^2 - 24*n + 8)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x))/(3*n - 2)); 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-e224ad3e3147ef52","steps":["D"],"edges":["e-3ebfbd075b124bf0"],"text":"A257622 \u2192 D"},"headline":"A257622 \u2192 D","chain":null}