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A337725
a(n) = (3*n+1)! * Sum_{k=0..n} 1 / (3*k+1)!.
6
1, 25, 5251, 3780721, 6487717237, 21798729916321, 126737815733490295, 1171057417377450325801, 16160592359808814496053801, 317652603424402057734433512457, 8567090714356123497097671830965291, 307592825008242258039794809418977808065
OFFSET
0,2
FORMULA
E.g.f.: (exp(3*x/2) - 2 * sin(Pi/6 - sqrt(3)*x/2)) / (3*exp(x/2) * (1 - x^3)) = x + 25*x^4/4! + 5251*x^7/7! + 3780721*x^10/10! + ...
a(n) = floor(c * (3*n+1)!), where c = (exp(3/2) + 2 * sin((3 * sqrt(3) - Pi) / 6))/(3 * sqrt(exp(1))) = A143820.
MATHEMATICA
Table[(3 n + 1)! Sum[1/(3 k + 1)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 1)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Pi/6 - Sqrt[3] x/2])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 1}], {n, 0, 11}]
Table[Floor[(Exp[3/2] + 2 Sin[(3 Sqrt[3] - Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 1)!], {n, 0, 11}]
PROG
(PARI) a(n) = (3*n+1)!*sum(k=0, n, 1/(3*k+1)!); \\ Michel Marcus, Sep 17 2020
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 17 2020
STATUS
approved