{"id":"n-d78f57cebf0818ff","revision":0,"first_index":0,"fingerprint":"f6-8a0079e810bae31e","name":"theta_plus_one_squared of basis_map_A013610 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":"ddc73a9fb60e522a6be1f296e1591560a4b7d0fda7c00ee29c7726300cd9f666","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-8a9c0145328414df","role":"sequence","name":"basis_map_A013610 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,        24;\n  2:       6,       120,        324;\n  3:      24,       624,       2916,       3456;\n  4:     120,      3696,      23004,      48384,       32400;\n  5:     720,     25056,     187920,     535680,      648000,      279936;\n  6:    5040,    192672,    1653696,    5754240,     9558000,     7558272,     2286144;\n  7:   40320,   1662336,   15830640,   63673344,   130410000,   143047296,    80015040,    17915904;\n  8:  362880,  15928704,  164917296,  741920256,  1770627600,  2414168064,  1888354944,   788299776,  136048896;\n  9: 3628800, 167996160, 1864210464, 9178320384, 24661778400, 39439903104, 38599255296, 22681534464, 7346640384, 1007769600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+24*n*x-324*x^2)*P(n-2,x) + (24*n*x^2+144*x^3)*d/dx P(n-2,x) + (2*n+1+24*x)*P(n-1,x) - 24*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 24*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 24*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 24*n*x - 324*x^2)*P[n - 2, x] + (24*n*x^2 + 144*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 24*x)*P[n - 1, x] - 24*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] = 2 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 24*n*x - 324*x^2)*v[i - 2] + (24*n*x^2 + 144*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 24*x)*v[i - 1] - 24*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":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-881cf76aa24e5971","e-bb3b5e75943ca428"],"text":"A137320 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A137320 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}