A251695 a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n.

1, 4, 175, 16250, 2348125, 463050000, 115966796875, 35253537343750, 12611991884765625, 5191587030710937500, 2417311348659677734375, 1256208098030090332031250, 720779749270420907470703125, 452589644988876542822265625000, 308707218248583408960223388671875

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..268

Formula

Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies:

(1) A(x) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4).

(2) A(x) = F(x*A(x)^4) where F(x) = exp(5*x*G(x)^4)/G(x) is the e.g.f. of A251665.

(3) A(x) = ( Series_Reversion( x*G(x)^4 / exp(20*x*G(x)^4) )/x )^(1/4).

E.g.f.: (-LambertW(-20*x)/(20*x))^(1/4) * (1 + LambertW(-20*x)/20). - Vaclav Kotesovec, Dec 07 2014

Example

E.g.f.: A(x) = 1 + 4*x + 175*x^2/2! + 16250*x^3/3! + 2348125*x^4/4! + 463050000*x^5/5! +...
such that A(x) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4),
where G(x) = 1 + x*G(x)^5 is the g.f. A002294:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^4) where
F(x) = 1 + 4*x + 47*x^2/2! + 1034*x^3/3! + 34349*x^4/4! + 1540480*x^5/5! +...
F(x) = exp( 5*x*G(x)^4 ) / G(x) is the e.g.f. of A251665.

Mathematica

Table[(3*n + 1)*(4*n + 1)^(n - 2)*5^n, {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

Prog

(PARI) {a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(G=1, A=1); for(i=0, n, G = 1 + x*G^5 +x*O(x^n));
A = ( serreverse( x*G^4 / exp(20*x*G^4) )/x )^(1/4); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [(3*n + 1)*(4*n + 1)^(n - 2)*5^n: n in [0..50]]; // G. C. Greubel, Nov 13 2017

Crossrefs

Cf. A251665, A002294.

Cf. Variants: A127670, A251693, A251694, A251696, A251697, A251698, A251699, A251700.

Sequence in context: A222921 A318216 A006433 * A343610 A113254 A210780

Adjacent sequences: A251692 A251693 A251694 * A251696 A251697 A251698

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2014

Status

approved