A337725 a(n) = (3*n+1)! * Sum_{k=0..n} 1 / (3*k+1)!.

1, 25, 5251, 3780721, 6487717237, 21798729916321, 126737815733490295, 1171057417377450325801, 16160592359808814496053801, 317652603424402057734433512457, 8567090714356123497097671830965291, 307592825008242258039794809418977808065

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..149

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.

D-finite with recurrence: 2260713*(3*n + 2)*(3*n + 4)*(n + 1)*a(n) + (146729799*n^3 + 880378794*n^2 + 1744454277*n + 1140478199)*a(n + 1) - 57*(2292408*n^3 + 20631672*n^2 + 61640304*n + 61226221)*a(n + 2) - 456*(46800*n^3 + 561600*n^2 + 2241200*n + 2963787)*a(n + 3) - 31616*(648*n^3 + 9720*n^2 + 48528*n + 80615)*a(n + 4) + 58368*(108*n^3 + 1944*n^2 + 11652*n + 23269)*a(n + 5) - 12288*(72*n^3 + 1512*n^2 + 10576*n + 24659)*a(n + 6) + 32768*a(n + 7) = 0. - Robert Israel, May 26 2026

Maple

f:= gfun:-rectoproc({2260713*(3*n + 2)*(3*n + 4)*(n + 1)*a(n) + (146729799*n^3 + 880378794*n^2 + 1744454277*n + 1140478199)*a(n + 1) - 57*(2292408*n^3 + 20631672*n^2 + 61640304*n + 61226221)*a(n + 2) - 456*(46800*n^3 + 561600*n^2 + 2241200*n + 2963787)*a(n + 3) - 31616*(648*n^3 + 9720*n^2 + 48528*n + 80615)*a(n + 4) + 58368*(108*n^3 + 1944*n^2 + 11652*n + 23269)*a(n + 5) - 12288*(72*n^3 + 1512*n^2 + 10576*n + 24659)*a(n + 6) + 32768*a(n + 7), a(0)=1, a(1)=25, a(2)=5251, a(3)=3780721, a(4)=6487717237, a(5)=21798729916321, a(6) = 126737815733490295}, a(n), remember):
map(f, [$0..30]); # Robert Israel, May 26 2026

Mathematica

Table[(3 n + 1)! Sum[1/(3 k + 1)!, {k, 0, n}], {n, 0, 11}]
(* Alternative: *)
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}]
(* Alternative: *)
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

Crossrefs

Cf. A000522, A051396, A051397, A087350, A100089, A143820, A330044, A337726, A337727, A337728, A337729, A337730.

Sequence in context: A185826 A178026 A012748 * A036512 A306380 A396493

Adjacent sequences: A337722 A337723 A337724 * A337726 A337727 A337728

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 17 2020

Status

approved