A246510 G.f.: Sum_{n>=0} x^n / (1-4*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k].

1, 5, 36, 305, 2821, 27690, 282699, 2967285, 31785786, 345815975, 3808549531, 42360017130, 474990254821, 5362633500755, 60897115958286, 695012481567465, 7966829676299139, 91674042449673960, 1058486539560201051, 12258669983923625475, 142359286920427682046, 1657287004720545992505

Offset: 0

Author

Paul D. Hanna, Aug 27 2014

Keywords

• nonn

Comments

a(n) == 1 (mod 3) iff n = A074939(k) for k>=0, where A074939 gives even numbers such that base 3 representation contains no 2.

a(n) == 2 (mod 3) iff n = A074938(k) for k>=0, where A074938 gives odd numbers such that base 3 representation contains no 2.

Examples

			G.f.: A(x) = 1 + 5*x + 36*x^2 + 305*x^3 + 2821*x^4 + 27690*x^5 +...
		

Crossrefs

Cf. A246509, A246056, A074938, A074939.

Programs

Mathematica

Table[Sum[3^k * Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 4^j,{j,0,n-2*k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Oct 04 2014 *)

PARI

/* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-4*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 3^k * x^k) * sum(k=0, m, binomial(m, k)^2 * 4^k * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* By a binomial identity: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 4^(m-k) * 3^k * x^k) * sum(k=0, m, binomial(m, k)^2 * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * 4^(k-j) * 3^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 4^(m-k) * sum(j=0, k, binomial(k, j)^2 * 3^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* Formula for a(n): */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 3^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 4^j))}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^(n-k) * 3^k * x^k].

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 4^(k-j) * 3^j * x^j.

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 4^(n-k) * Sum_{j=0..k} C(k,j)^2 * 3^j * x^j.

a(n) = Sum_{k=0..[n/2]} 3^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 4^j.

a(n) ~ sqrt(36 + 29*sqrt(3) + 3*sqrt(423 + 232*sqrt(3))) * (9/2 + sqrt(3) + 3/2*sqrt(9 + 4*sqrt(3)))^n / (8*Pi*n). - Vaclav Kotesovec, Oct 04 2014