A180718 G.f.: exp( Sum_{n>=0} [ Sum_{k=0..n} C(n,k)^2*x^k ]^2* x^n/n ).

1, 1, 3, 8, 25, 80, 271, 952, 3443, 12758, 48212, 185283, 722227, 2849955, 11366379, 45757142, 185726603, 759401542, 3125472832, 12939604503, 53856950922, 225250407802, 946253665230, 3991221520996, 16897320866269, 71782331694315

Offset: 0

Author

Paul D. Hanna, Sep 24 2010

Keywords

• nonn

Comments

Compare the g.f. of this sequence to the g.f.s:

. exp( Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k]*x^n/n ) = (G(x)-1)/x where G(x) = g.f. of A004148.

. exp( Sum_{n>=0} [Sum_{k=0..n} C(n,k)*x^k]^2*x^n/n ) = 1/(1-x*(1+x)^2).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 80*x^5 + 271*x^6 +...
The logarithm (A180719) begins:
log(A(x)) = x + 5*x^2/2 + 16*x^3/3 + 61*x^4/4 + 226*x^5/5 + 884*x^6/6 + 3543*x^7/7 + 14429*x^8/8 +...
which equals the sum of the series:
log(A(x)) = (1 + x)^2*x
+ (1 + 4*x + x^2)^2*x^2/2
+ (1 + 9*x + 9*x^2 + x^3)^2*x^3/3
+ (1 + 16*x + 36*x^2 + 16*x^3 + x^4)^2*x^4/4
+ (1 + 25*x + 100*x^2 + 100*x^3 + 25*x^4 + x^5)^2*x^5/5
+ (1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)^2*x^6/6 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..900

Crossrefs

Cf. A180719, A004148.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^2*x^k)^2*x^m/m)+x*O(x^n)),n)}