A183166 G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k]^n*x^n.

1, 1, 2, 9, 46, 343, 3025, 32811, 417348, 6106921, 102307571, 1918139824, 40190540565, 928661958828, 23551552524966, 651213150740841, 19523328447786346, 631923020784069573, 21984209405892842663, 819109566359501449734, 32576039720255480451008, 1378639634715738629523321

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 46*x^4 + 343*x^5 + 3025*x^6 +...
equals the sum of the series:
A(x) = 1 + (1+x)*x + (1 + 2^2*x + x^2)^2*x^2 +
+ (1 + 3^2*x + 3^2*x^2 + x^3)^3*x^3
+ (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^4*x^4
+ (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^5*x^5
+ (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 +...
The g.f. can also be expressed as:
A(x) = 1 + x*(1-x)^3*(1 + 2^2*x + 3^2*x^2 + 4^2*x^3 + 5^2*x^4 +...)
+ x^2*(1-x)^10*(1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 +...)^2
+ x^3*(1-x)^21*(1 + 4^2*x + 10^2*x^2 + 20^2*x^3 + 35^2*x^4 +...)^3
+ x^4*(1-x)^36*(1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 +...)^4
+ x^5*(1-x)^55*(1 + 6^2*x + 21^2*x^2 + 56^2*x^3 + 126^2*x^4 +...)^5 +...

Prog

(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)}
for(n=0, 30, print1(a(n), ", "))
(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)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A183165, A184355, A184356.

Sequence in context: A365855 A134091 A219614 * A032331 A049371 A306725

Adjacent sequences: A183163 A183164 A183165 * A183167 A183168 A183169

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2010

Status

approved