A226705 G.f.: 1 / (1 + 12*x*G(x)^4 - 16*x*G^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

1, 4, 48, 600, 7856, 105684, 1447392, 20075416, 281086416, 3964453368, 56240518128, 801624722232, 11470976280960, 164691196943212, 2371222443727584, 34224696393237360, 495036708728067088, 7173892793100898728, 104135761805147016096, 1513892435551302963792

Offset: 0

Author

Paul D. Hanna, Jun 15 2013

Keywords

• nonn

Examples

			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).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A226706, A183160, A147855, A002295.

Cf. A226733, A226751.

Cf. A004355, A079679.

Programs

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 *)

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), ", "))

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

From Seiichi Manyama, Aug 05 2025: (Start)

a(n) = [x^n] 1/((1+2*x) * (1-x)^(5*n+1)).

a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(6*n+1,k).

a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+k,k). (End)

From Seiichi Manyama, Aug 14 2025: (Start)

a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).

G.f.: G(x)^2/((-2+3*G(x)) * (6-5*G(x))) where G(x) = 1+x*G(x)^6 is the g.f. of A002295. (End)

G.f.: B(x)^2/(1 + 4*(B(x)-1)/3), where B(x) is the g.f. of A004355. - Seiichi Manyama, Aug 15 2025