{"id":"n-d208ccf34f843ce5","revision":0,"first_index":0,"fingerprint":"f6-44883e8bcc0275dc","name":"theta_plus_one of basis_map_A013611 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":"abd5515cc05c5e3956d4a9371720e43f047c174d5e64403ecef3ea99d0e54574","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-07810086d0e6802d","role":"sequence","name":"basis_map_A013611 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,        16;\n  2:       6,        80,        192;\n  3:      24,       416,       1728,       2048;\n  4:     120,      2464,      13632,      28672,       20480;\n  5:     720,     16704,     111360,     317440,      409600,      196608;\n  6:    5040,    128448,     979968,    3409920,     6041600,     5308416,     1835008;\n  7:   40320,   1108224,    9381120,   37732352,    82432000,   100466688,    64225280,    16777216;\n  8:  362880,  10619136,   97728768,  439656448,  1119211520,  1695547392,  1515716608,   738197504,  150994944;\n  9: 3628800, 111997440, 1104717312, 5439004672, 15588679680, 27699904512, 30982275072, 21239955456, 8153726976, 1342177280;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+16*n*x+64*x^2)*P(n-2,x) + (2*n+1+16*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 16*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 16*n*x + 64*x^2)*P[n - 2, x] + (2*n + 1 + 16*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 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 16*n*x + 64*x^2)*v[i - 2] + (2*n + 1 + 16*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":["basis A013611","\u03b8+1"],"edges":["e-40669049634a7e59","e-0dbdde53893caa52"],"text":"A137320 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A137320 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}