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G.f.: Sum_{n>=0} [Sum_{k>=0} C(n+k-1,k)^2*x^k]^n * x^n.
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%I #12 Nov 24 2014 22:52:02

%S 1,1,2,10,63,521,5295,64048,907199,14717173,270429934,5561319631,

%T 126824201866,3183127838869,87328494060529,2604069098659922,

%U 83975195990867113,2915521294244073351,108553405498985038390,4319110373993534510736,183057722816741327269600

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

%H Paul D. Hanna, <a href="/A183165/b183165.txt">Table of n, a(n) for n = 0..100</a>

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

%e G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 63*x^4 + 521*x^5 + 5295*x^6 +...

%e which equals the sum of the series:

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

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

%e + (1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 + 21^2*x^5 +...)^3*x^3

%e + (1 + 4^2*x + 10^2*x^2 + 20^2*x^3 + 35^2*x^4 + 56^2*x^5 +...)^4*x^4

%e + (1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 + 126^2*x^5 +...)^5*x^5

%e + (1 + 6^2*x + 21^2*x^2 + 56^2*x^3 + 126^2*x^4 + 252^2*x^5 +...)^6*x^6 +...

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

%e A(x) = 1 + x/(1-x) + (1+x)^2*x^2/(1-x)^6

%e + (1 + 2^2*x + x^2)^3*x^3/(1-x)^15

%e + (1 + 3^2*x + 3^2*x^2 + x^3)^4*x^4/(1-x)^28

%e + (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^5*x^5/(1-x)^45

%e + (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^6*x^6/(1-x)^66

%e + (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^7*x^7/(1-x)^91 +...

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

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

%Y Cf. A183166, A184355, A184356.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 30 2010