{"id":"n-8601e37f53e74b20","revision":0,"first_index":0,"fingerprint":"f6-23668f2803585b2c","name":"theta_plus_one_squared of Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, 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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":"a61b0ff71c4a32bfc6001ec4eeb01a7957987d2cfc2252db9d447599fc2ebd67","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-633aaaf6fe98dc51","role":"sequence","name":"Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.","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: 1,     8;\n  2: 1,    40,      36;\n  3: 1,   120,     648,       128;\n  4: 1,   296,    4644,      7424,        400;\n  5: 1,   664,   23256,    112256,      71600,        1152;\n  6: 1,  1416,   96660,   1027840,    2112000,      625536,        3136;\n  7: 1,  2936,  360072,   7204480,   33044000,    34166016,     5130496,       8192;\n  8: 1,  5992, 1251972,  42900736,  367846000,   871163136,   500950912,   40271872,     20736;\n  9: 1, 12120, 4160376, 229471872, 3313942800, 14761547136, 20107153920, 6865207296, 306001152, 51200;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = -((32*n^2-32*n-24)*x-(288*n^3-496*n^2+56*n+60)*x^2)*P(n-2,x) - ((4*n+2)*x-(32*n^2+32*n+16)*x^2-(128*n^3-640*n^2+128*n+80)*x^3)*d/dx P(n-2,x) - ((2*n+1)*x^2-32*n^2*x^3+(128*n^2-32*n-16)*x^4)*d^2/dx^2 P(n-2,x) + (2*n-1+(32*n^2-32*n+8)*x)*P(n-1,x) + (4*n*x-(32*n^2-8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((32*n^2 - 32*n - 24)*x - (288*n^3 - 496*n^2 + 56*n + 60)*x^2)*P[n - 2, x] - ((4*n + 2)*x - (32*n^2 + 32*n + 16)*x^2 - (128*n^3 - 640*n^2 + 128*n + 80)*x^3)*D[P[n - 2, x], x] - ((2*n + 1)*x^2 - 32*n^2*x^3 + (128*n^2 - 32*n - 16)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n - 1 + (32*n^2 - 32*n + 8)*x)*P[n - 1, x] + (4*n*x - (32*n^2 - 8)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-((32*n^2 - 32*n - 24)*x - (288*n^3 - 496*n^2 + 56*n + 60)*x^2)*v[i - 2] - ((4*n + 2)*x - (32*n^2 + 32*n + 16)*x^2 - (128*n^3 - 640*n^2 + 128*n + 80)*x^3)*deriv(v[i - 2], x) - ((2*n + 1)*x^2 - 32*n^2*x^3 + (128*n^2 - 32*n - 16)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n - 1 + (32*n^2 - 32*n + 8)*x)*v[i - 1] + (4*n*x - (32*n^2 - 8)*x^2)*deriv(v[i - 1], x))/(2*n - 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-633aaaf6fe98dc51","steps":["(\u03b8+1)\u00b2"],"edges":["e-c43f5d4b8ff0f427"],"text":"A142963 \u2192 (\u03b8+1)\u00b2"},"headline":"A142963 \u2192 (\u03b8+1)\u00b2","chain":null}