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G.f.: Sum_{n>=0} x^n / (1-3*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, 4, 26, 200, 1694, 15224, 141972, 1359120, 13264246, 131384536, 1316769196, 13323636208, 135885868780, 1395157192624, 14406117404584, 149489132177440, 1557898906160806, 16297193704008856, 171058624529373116, 1800860588158214960, 19010179617892702404, 201164103801453466896

0,2

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 * 3^(n-k) * 4^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 * 3^(k-j) * 4^j * x^j.

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

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

G.f.: A(x) = 1 + 4*x + 26*x^2 + 200*x^3 + 1694*x^4 + 15224*x^5 +...

(PARI) /* By definition: */

{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-3*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 * 3^(m-k) * 4^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 * 3^(k-j) * 4^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 * 3^(m-k) * sum(j=0, k, binomial(k, j)^2 * 4^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, 4^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 3^j))}

for(n=0, 25, print1(a(n), ", "))

Cf. A246510, A246423.

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Paul D. Hanna, Aug 27 2014

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