A183166 - OEIS

%I #13 Nov 24 2014 22:41:19

%S 1,1,2,9,46,343,3025,32811,417348,6106921,102307571,1918139824,

%T 40190540565,928661958828,23551552524966,651213150740841,

%U 19523328447786346,631923020784069573,21984209405892842663,819109566359501449734,32576039720255480451008,1378639634715738629523321

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

%H Paul D. Hanna, <a href="/A183166/b183166.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: Sum_{n>=0} x^n * (1-x)^(2*n^2 + n) * [Sum_{k>=0} C(n+k,k)^2 * x^k]^n.

%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 46*x^4 + 343*x^5 + 3025*x^6 +...

%e equals the sum of the series:

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

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

%e + (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^4*x^4

%e + (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^5*x^5

%e + (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^6*x^6 +...

%e The g.f. can also be expressed as:

%e A(x) = 1 + x*(1-x)^3*(1 + 2^2*x + 3^2*x^2 + 4^2*x^3 + 5^2*x^4 +...)

%e + x^2*(1-x)^10*(1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 +...)^2

%e + x^3*(1-x)^21*(1 + 4^2*x + 10^2*x^2 + 20^2*x^3 + 35^2*x^4 +...)^3

%e + x^4*(1-x)^36*(1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 +...)^4

%e + x^5*(1-x)^55*(1 + 6^2*x + 21^2*x^2 + 56^2*x^3 + 126^2*x^4 +...)^5 +...

%o (PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(sum(m=0, n, sum(k=0, m, binomial(m, k)^2*X^k)^m*x^m) +x*O(x^n), n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*(1-x+x*O(x^n))^(2*m^2+m)*sum(k=0, n-m+1, binomial(m+k, k)^2*x^k+x*O(x^n))^m), n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A183165, A184355, A184356.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 30 2010