{"id":"n-f95e6c0740980414","revision":0,"first_index":0,"fingerprint":"f6-8fbe907eb3154833","name":"theta_plus_one_squared of reversal 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":"047ada48bedc89cdbb58c05223b0eb18eec1746389b19101de75c6168b43cac6","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-3df3926aefc1fd76","role":"sequence","name":"reversal 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;\n  2:   4,     8;\n  3:   8,    48,     36;\n  4:  16,   192,    396,     192;\n  5:  32,   640,   2520,    3200,     1200;\n  6:  64,  1920,  12240,   28800,    27400,     8640;\n  7: 128,  5376,  50400,  188160,   324800,   254016,    70560;\n  8: 256, 14336, 185472, 1003520,  2707600,  3782016,  2561328,   645120;\n  9: 512, 36864, 628992, 4644864, 17959200, 38755584, 46304608, 28053504, 6531840;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(4*n-4+(8*n^2-24*n+16)*x-(9*n^3-45*n^2+72*n-36)*x^2)*P(n-2,x) + ((8*n^2-24*n+16)*x^2+(4*n^3-20*n^2+32*n-16)*x^3)*d/dx P(n-2,x) + (4*n-6+(4*n^2-12*n+8)*x)*P(n-1,x) - (4*n^2-12*n+8)*x^2*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 4 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 4 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 4 + (8*n^2 - 24*n + 16)*x - (9*n^3 - 45*n^2 + 72*n - 36)*x^2)*P[n - 2, x] + ((8*n^2 - 24*n + 16)*x^2 + (4*n^3 - 20*n^2 + 32*n - 16)*x^3)*D[P[n - 2, x], x] + (4*n - 6 + (4*n^2 - 12*n + 8)*x)*P[n - 1, x] - (4*n^2 - 12*n + 8)*x^2*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 4 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 4 + (8*n^2 - 24*n + 16)*x - (9*n^3 - 45*n^2 + 72*n - 36)*x^2)*v[i - 2] + ((8*n^2 - 24*n + 16)*x^2 + (4*n^3 - 20*n^2 + 32*n - 16)*x^3)*deriv(v[i - 2], x) + (4*n - 6 + (4*n^2 - 12*n + 8)*x)*v[i - 1] - (4*n^2 - 12*n + 8)*x^2*deriv(v[i - 1], x))/(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-60c7e36cbe887b7e","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-a28ae8b138b0cf1e","e-ddc0923472cbb707"],"text":"A137320 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A137320 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}