OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k=0..n} C(2*k, n-k) * C(6*n-2*k, k).
a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(5*n-2*k, k).
a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(4*n-2*k, k).
a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(3*n-2*k, k).
a(n) = Sum_{k=0..n} C(4*n+2*k, n-k) * C(2*n-2*k, k).
a(n) = Sum_{k=0..n} C(5*n+2*k, n-k) * C(n-2*k, k).
a(n) = Sum_{k=0..n} C(6*n+2*k, n-k) * C(-2*k, k).
Self-convolution of A226706.
G.f.: 1 / (1 - 4*x*G(x)^4 - 16*x^2*G(x)^10) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
a(n) ~ 2^(6*n-2)*3^(6*n+3/2)/(5^(5*n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 16 2013
EXAMPLE
G.f.: A(x) = 1 + 4*x + 48*x^2 + 600*x^3 + 7856*x^4 + 105684*x^5 +...
A related series is G(x) = 1 + x*G(x)^6, where
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
G(x)^4 = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +...
G(x)^5 = 1 + 5*x + 40*x^2 + 385*x^3 + 4095*x^4 + 46376*x^5 +...
such that A(x) = 1/(1 + 12*x*G(x)^4 - 16*x*G^5).
MATHEMATICA
Table[Sum[Binomial[3*n+2*k, n-k]*Binomial[3*n-2*k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 16 2013 *)
PROG
(PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^6+x*O(x^n)); polcoeff(1/(1+12*x*G^4-16*x*G^5), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^6+x*O(x^n)); polcoeff(1/(1-4*x*G^4-16*x^2*G^10), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(6*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, binomial(3*n +2*k, n-k)*binomial(3*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, binomial(6*n +2*k, n-k)*binomial(-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 15 2013
STATUS
approved