{"id":"n-c3c114a9c72eed60","revision":1,"first_index":0,"fingerprint":"f6-492960c5c260ecd7","name":"theta_plus_one_squared 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":"3672c463821c5fcd8ba3c00ec920a5c58545dc1709415ed5cf33c6c4b3c454a1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-60c7e36cbe887b7e","role":"sequence","name":"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: 0,      8;\n  2: 0,      8,      36;\n  3: 0,     16,     108,      128;\n  4: 0,     48,     396,      768,      400;\n  5: 0,    192,    1800,     4480,     4000,     1152;\n  6: 0,    960,    9864,    28800,    34000,    17280,     3136;\n  7: 0,   5760,   63504,   207872,   294000,   201600,    65856,    8192;\n  8: 0,  40320,  470448,  1680896,  2707600,  2257920,  1009792,  229376,  20736;\n  9: 0, 322560, 3945024, 15119872, 26913600, 25861248, 14224896, 4472832, 746496, 51200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-4*n+4+(8*n-16)*x-36*x^2)*P(n-2,x) + ((8*n-16)*x^2+16*x^3)*d/dx P(n-2,x) + (2*n-3+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 4 + (8*n - 16)*x - 36*x^2)*P[n - 2, x] + ((8*n - 16)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (2*n - 3 + 8*x)*P[n - 1, x] - 8*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 4 + (8*n - 16)*x - 36*x^2)*v[i - 2] + ((8*n - 16)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (2*n - 3 + 8*x)*v[i - 1] - 8*x^2*deriv(v[i - 1], x)); 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":["(\u03b8+1)\u00b2"],"edges":["e-ab1c8ef10bffa507"],"text":"A137320 \u2192 (\u03b8+1)\u00b2"},"headline":"A137320 \u2192 (\u03b8+1)\u00b2","chain":null}