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A121689
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G.f.: A(x) = Sum_{k>=0} x^k * (1+x)^(k^2).
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1
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1, 1, 2, 5, 16, 57, 231, 1023, 4926, 25483, 140601, 822422, 5074015, 32881868, 223027542, 1578435549, 11625317128, 88894615929, 704269188135, 5770209550496, 48810504348082, 425650324975153, 3821377057170313
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| a(n) = Sum_{k=0..n} C(k^2,n-k).
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 24 2010: (Start)
Let q = (1+x), then g.f. A(x) equals the continued fraction:
A(x) = 1/(1- q*x/(1- (q^3-q)*x/(1- q^5*x/(1- (q^7-q^3)*x/(1- q^9*x/(1- (q^11-q^5)*x/(1- q^13*x/(1- (q^15-q^7)*x/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
(End)
G.f.: A(x) = Sum_{n>=0} x^n*(1+x)^n*Product_{k=1..n} (1-x*(1+x)^(4*k-3))/(1-x*(1+x)^(4*k-1)) due to a q-series identity. [From Paul D. Hanna (pauldhanna(AT)juno.com), May 08 2010]
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PROG
| (PARI) a(n)=sum(k=0, n, binomial(k^2, n-k))
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*(1+x)^m*prod(k=1, m, (1-x*(1+x)^(4*k-3))/(1-x*(1+x)^(4*k-1) +x*O(x^n)))), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), May 08 2010]
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CROSSREFS
| Sequence in context: A197158 A188314 A114296 * A192635 A009225 A157612
Adjacent sequences: A121686 A121687 A121688 * A121690 A121691 A121692
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 15 2006
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