A251663 E.g.f.: exp( 3*x*G(x)^2 ) / G(x), where G(x) = 1 + x*G(x)^3 is the g.f. A001764.

1, 2, 11, 120, 2061, 48918, 1487151, 55188108, 2419385625, 122367255498, 7014349322739, 449405251066368, 31826192109186789, 2468711973793223070, 208159999898813165079, 18957203713618483723092, 1854424578467714146269489, 193922780991931737971748882, 21588348501840566333913576795

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

Let G(x) = 1 + x*G(x)^3 be the g.f. of A001764, then the e.g.f. A(x) of this sequence satisfies:

(1) A'(x)/A(x) = G(x)^2 + G'(x)/G(x).

(2) A(x) = F(x/A(x)^2) where F(x) is the e.g.f. of A251693.

(3) A(x) = Sum_{n>=0} A251693(n)*(x/A(x)^2)^n/n! where A251693(n) = (n+1) * (2*n+1)^(n-2) * 3^n.

(4) [x^n/n!] A(x)^(2*n+1) = (n+1) * (2*n+1)^(n-1) * 3^n.

a(n) = Sum_{k=0..n} 3^k * n!/k! * binomial(3*n-k-2, n-k) * (2*k-1)/(2*n-1) for n>=0.

Recurrence: 2*(2*n-1)*(9*n^2 - 30*n + 19)*a(n) = 3*(81*n^4 - 432*n^3 + 756*n^2 - 393*n - 88)*a(n-1) - 27*(9*n^2 - 12*n - 2)*a(n-2). - Vaclav Kotesovec, Dec 07 2014

a(n) ~ 3^(3*n-3/2) * n^(n-1) / (2^(2*n-1/2) * exp(n-1)). - Vaclav Kotesovec, Dec 07 2014

Example

E.g.f.: A(x) = 1 + 2*x + 11*x^2/2! + 120*x^3/3! + 2061*x^4/4! + 48918*x^5/5! +...
such that A(x) = exp(3*x*G(x)^2) / G(x)
where G(x) = 1 + x*G(x)^3 is the g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
The e.g.f. satisfies:
A(x) = 1 + 2*x/A(x)^2 + 27*x^2/(2!*A(x)^4) + 756*x^3/(3!*A(x)^6) + 32805*x^4/(4!*A(x)^8) + 1940598*x^5/(5!*A(x)^10) + 145746783*x^6/(6!*A(x)^12) + 13286025000*x^7/(7!*A(x)^14) +...+ (n+1)*(2*n+1)^(n-2)*3^n * x^n/(n!*A(x)^(2*n)) +...

Mathematica

Table[Sum[3^k * n!/k! * Binomial[3*n-k-2, n-k] * (2*k-1)/(2*n-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 07 2014 *)

Prog

(PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^3 +x*O(x^n)); n!*polcoeff(exp(3*x*G^2)/G, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, 3^k * n!/k! * binomial(3*n-k-2, n-k) * (2*k-1)/(2*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A251573, A251693, A001764.

Cf. Variants: A243953, A251664, A251665, A251666, A251667, A251668, A251669, A251670.

Sequence in context: A197993 A057076 A346650 * A118794 A222879 A354978

Adjacent sequences: A251660 A251661 A251662 * A251664 A251665 A251666

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2014

Status

approved