{"id":"n-385c0b04bd494306","revision":0,"first_index":0,"fingerprint":"f6-713e6aebbd821571","name":"theta_plus_one of basis_map_A007318 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. 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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"869a5a5d1aa8ecf897248f7bdcf56271ce628830dab04ce402d773fa288b65e7","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-7a2a38f140d4a893","role":"sequence","name":"basis_map_A007318 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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:       1;\n  1:       2,        4;\n  2:       6,       20,       12;\n  3:      24,      104,      108,       32;\n  4:     120,      616,      852,      448,       80;\n  5:     720,     4176,     6960,     4960,     1600,      192;\n  6:    5040,    32112,    61248,    53280,    23600,     5184,     448;\n  7:   40320,   277056,   586320,   589568,   322000,    98112,   15680,    1024;\n  8:  362880,  2654784,  6108048,  6869632,  4371920,  1655808,  370048,   45056,   2304;\n  9: 3628800, 27999360, 69044832, 84984448, 60893280, 27050688, 7564032, 1296384, 124416, 5120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+4*n*x+4*x^2)*P(n-2,x) + (2*n+1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n*x + 4*x^2)*P[n - 2, x] + (2*n + 1 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n*x + 4*x^2)*v[i - 2] + (2*n + 1 + 4*x)*v[i - 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-60c7e36cbe887b7e","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-01af55cf26557a61","e-b9548bd5dbe25c97"],"text":"A137320 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A137320 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}