A380724 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^3) / (1 - x*A(x)^3).

1, 2, 29, 862, 39461, 2454296, 193406953, 18475039808, 2075062993865, 268013104242688, 39139481641977461, 6377306725457207552, 1147019426037344539501, 225728971809041691392000, 48248339461852786811399489, 11131014193619108036340637696, 2756799306857952163745291500433

Offset

0,2

Links

Table of n, a(n) for n=0..16.

Formula

a(n) = n! * Sum_{k=0..n} (3*n+1)^(k-1) * binomial(4*n-k,n-k)/k!.

a(n) ~ sqrt(2 + 11/sqrt(37)) * 2^n * 3^(4*n - 3/2) * exp((-3*(sqrt(37)-5)*n-sqrt(37)+7)/6) * n^(n-1) / (49*sqrt(37)-283)^n. - Vaclav Kotesovec, Jan 31 2026

Mathematica

Table[n! * Sum[(3*n+1)^(k-1) * Binomial[4*n-k, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 31 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, (3*n+1)^(k-1)*binomial(4*n-k, n-k)/k!);

Crossrefs

Cf. A377831, A380723.

Cf. A373324, A377889.

Cf. A360609.

Sequence in context: A245252 A090251 A087281 * A389722 A024234 A367551

Adjacent sequences: A380721 A380722 A380723 * A380725 A380726 A380727

Keyword

nonn

Author

Seiichi Manyama, Jan 30 2025

Status

approved