A180718 - OEIS

%I #4 Jul 31 2014 09:48:44

%S 1,1,3,8,25,80,271,952,3443,12758,48212,185283,722227,2849955,

%T 11366379,45757142,185726603,759401542,3125472832,12939604503,

%U 53856950922,225250407802,946253665230,3991221520996,16897320866269,71782331694315

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

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

%C . 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.

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

%H Vaclav Kotesovec, <a href="/A180718/b180718.txt">Table of n, a(n) for n = 0..900</a>

%e G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 80*x^5 + 271*x^6 +...

%e The logarithm (A180719) begins:

%e 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 +...

%e which equals the sum of the series:

%e log(A(x)) = (1 + x)^2*x

%e + (1 + 4*x + x^2)^2*x^2/2

%e + (1 + 9*x + 9*x^2 + x^3)^2*x^3/3

%e + (1 + 16*x + 36*x^2 + 16*x^3 + x^4)^2*x^4/4

%e + (1 + 25*x + 100*x^2 + 100*x^3 + 25*x^4 + x^5)^2*x^5/5

%e + (1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)^2*x^6/6 +...

%o (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)}

%Y Cf. A180719, A004148.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 24 2010