A251694 a(n) = (2*n+1) * (3*n+1)^(n-2) * 4^n.

1, 3, 80, 4480, 389376, 46137344, 6939332608, 1266556600320, 272000000000000, 67204714785144832, 18780742521990414336, 5857307089452073484288, 2016866466756967373209600, 759982437118771200000000000, 311070869724989874190180941824, 137440072511222468264810285891584

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..283

Formula

Let G(x) = 1 + x*G(x)^4 be the g.f. of A002293, then the e.g.f. A(x) of this sequence satisfies:

(1) A(x) = exp( 4*x*A(x)^3 * G(x*A(x)^3)^3 ) / G(x*A(x)^3).

(2) A(x) = F(x*A(x)^3) where F(x) = exp(4*x*G(x)^3)/G(x) is the e.g.f. of A251664.

(3) A(x) = ( Series_Reversion( x*G(x)^3 / exp(12*x*G(x)^3) )/x )^(1/3).

E.g.f.: (-LambertW(-12*x)/(12*x))^(1/3) * (1 + LambertW(-12*x)/12). - Vaclav Kotesovec, Dec 07 2014

Example

E.g.f.: A(x) = 1 + 3*x + 80*x^2/2! + 4480*x^3/3! + 389376*x^4/4! + 46137344*x^5/5! +...
such that A(x) = exp( 4*x*A(x)^3 * G(x*A(x)^3)^3 ) / G(x*A(x)^3),
where G(x) = 1 + x*G(x)^4 is the g.f. A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^3) where
F(x) = 1 + 3*x + 26*x^2/2! + 430*x^3/3! + 10872*x^4/4! + 373664*x^5/5! +...
F(x) = exp( 4*x*G(x)^3 ) / G(x) is the e.g.f. of A251664.

Mathematica

Table[(2*n + 1)*(3*n + 1)^(n - 2)*4^n, {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

Prog

(PARI) {a(n) = (2*n+1) * (3*n+1)^(n-2) * 4^n}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(G=1, A=1); for(i=0, n, G = 1 + x*G^4 +x*O(x^n));
A = ( serreverse( x*G^3 / exp(12*x*G^3) )/x )^(1/3); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [ (2*n + 1)*(3*n + 1)^(n - 2)*4^n: n in [0..50]]; // G. C. Greubel, Nov 13 2017

Crossrefs

Cf. A251664, A002293.

Cf. Variants: A127670, A251693, A251695, A251696, A251697, A251698, A251699, A251700.

Sequence in context: A197248 A183265 A233124 * A355626 A274567 A137994

Adjacent sequences: A251691 A251692 A251693 * A251695 A251696 A251697

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2014

Status

approved