A147855 G.f.: 1 / (1 + 4*x*G(x)^2 - 7*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

1, 3, 22, 174, 1444, 12323, 107104, 942952, 8381596, 75053100, 676017962, 6118171326, 55591175956, 506805088026, 4633571685968, 42468065811884, 390071875757852, 3589637747968964, 33089300640166360, 305476314574338648, 2823932709938708824, 26137341654281261347

Offset: 0

Author

Paul D. Hanna, Jun 16 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 3*x + 22*x^2 + 174*x^3 + 1444*x^4 + 12323*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 4*x*G(x)^2 - 7*x*G(x)^3).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A226705, A002293.

Cf. A026641, A183160, A371753.

Cf. A005810, A078995, A226733, A226761, A385605, A386811.

Programs

Mathematica

Table[Sum[Binomial[2*n+k,n-k]*Binomial[2*n-k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 16 2013 *)

PARI

{a(n)=sum(k=0, n, binomial(2*n+k, n-k)*binomial(2*n-k, k))}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n)=sum(k=0, n, binomial(k, n-k)*binomial(4*n-k, k))}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+4*x*G^2-7*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-3*x*G^2-7*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} C(k, n-k) * C(4*n-k, k).

a(n) = Sum_{k=0..n} C(n+k, n-k) * C(3*n-k, k).

a(n) = Sum_{k=0..n} C(2*n+k, n-k) * C(2*n-k, k).

a(n) = Sum_{k=0..n} C(3*n+k, n-k) * C(n-k, k).

a(n) = Sum_{k=0..n} C(4*n+k, n-k) * C(-k, k).

G.f.: 1 / (1 - 3*x*G(x)^2 - 7*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

a(n) ~ 2^(8*n+5/2)/(5*sqrt(Pi*n)*3^(3*n+1/2)). - Vaclav Kotesovec, Jun 16 2013

From Seiichi Manyama, Apr 05 2024: (Start)

a(n) = Sum_{k=0..floor(n/2)} binomial(4*n-2*k-1,n-2*k).

a(n) = [x^n] 1/((1-x^2) * (1-x)^(3*n)). (End)

From Seiichi Manyama, Aug 05 2025: (Start)

a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+1,k).

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

From Seiichi Manyama, Aug 14 2025: (Start)

a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

G.f.: G(x)^2/((-1+2*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)

G.f.: B(x)^2/(1 + 5*(B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025