A251583 a(n) = 3^(n-1) * (n+1)^(n-3) * (n+3).
1, 1, 5, 54, 945, 23328, 750141, 29859840, 1420541793, 78732000000, 4986357828309, 355459848339456, 28178328756432465, 2459548529521606656, 234438580086767578125, 24233149581890213117952, 2700277512299794365456321, 322689729227525728790446080, 41170357602396483760424637477, 5585797616762880000000000000000
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 54*x^3/3! + 945*x^4/4! + 23328*x^5/5! +... such that A(x) = exp( 3*x*A(x) * G(x*A(x))^2 ) / G(x*A(x))^2 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 +... RELATED SERIES. Note that A(x) = F(x*A(x)) where F(x) = exp(3*x*G(x)^2)/G(x)^2, F(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 261*x^4/4! + 4833*x^5/5! +... is the e.g.f. of A251573.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..330
Crossrefs
Programs
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Mathematica
Table[3^(n - 1)*(n + 1)^(n - 3)*(n + 3), {n, 0, 20}] (* G. C. Greubel, Nov 12 2017 *) -
PARI
{a(n) = 3^(n-1) * (n+1)^(n-3) * (n+3)} for(n=0,20,print1(a(n),", ")) -
PARI
{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^3 +x*O(x^n)); for(i=1,n, A = exp(3*x*A * subst(G^2,x,x*A) ) / subst(G^2,x,x*A) ); n!*polcoeff(A, n)} for(n=0, 20, print1(a(n), ", "))
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) = exp( 3*x*A(x) * G(x*A(x))^2 ) / G(x*A(x))^2.
(2) A(x) = F(x*A(x)) where F(x) = exp(3*x*G(x)^2)/G(x)^2 is the e.g.f. of A251573.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251573.
E.g.f.: -LambertW(-3*x) * (3 + LambertW(-3*x))^2 / (27*x). - Vaclav Kotesovec, Dec 07 2014