A251586 a(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296).

1, 1, 8, 156, 5160, 245976, 15450912, 1209613824, 113666333184, 12479546880000, 1568823886181376, 222308476014034944, 35069155573323036672, 6096327654732137496576, 1158040133351856000000000, 238674982804212474577944576, 53050036437721656891731017728, 12649916782354997981599305302016

Offset: 0

Author

Paul D. Hanna, Dec 06 2014

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 156*x^3/3! + 5160*x^4/4! + 245976*x^5/5! +...
such that A(x) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(x))^5
where G(x) = 1 + x*G(x)^6 is the g.f. of A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5,
F(x) = 1 + x + 6*x^2/2! + 96*x^3/3! + 2736*x^4/4! + 115056*x^5/5! +...
is the e.g.f. of A251576.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..306

Crossrefs

Cf. A251576, A002295.

Cf. Variants: A251583, A251584, A251585, A251587, A251588, A251589, A251590.

Programs

Magma

[6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Mathematica

Table[6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

PARI

{a(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296)}
for(n=0,20,print1(a(n),", "))

PARI

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^6 +x*O(x^n));
for(i=1,n, A = exp(6*x*A * subst(G^5,x,x*A) ) / subst(G^5,x,x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:

(1) A(x) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(x))^5.

(2) A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5 is the e.g.f. of A251576.

(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251576.

E.g.f.: -LambertW(-6*x) * (6 + LambertW(-6*x))^5 / (x*6^6). - Vaclav Kotesovec, Dec 07 2014