cp's OEIS Frontend

A246509 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].

%I A246509 #7 May 05 2025 20:22:45
%S A246509 1,4,26,200,1694,15224,141972,1359120,13264246,131384536,1316769196,
%T A246509 13323636208,135885868780,1395157192624,14406117404584,
%U A246509 149489132177440,1557898906160806,16297193704008856,171058624529373116,1800860588158214960,19010179617892702404,201164103801453466896
%N A246509 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].
%F A246509 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].
%F A246509 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.
%F A246509 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.
%F A246509 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.
%e A246509 G.f.: A(x) = 1 + 4*x + 26*x^2 + 200*x^3 + 1694*x^4 + 15224*x^5 +...
%o A246509 (PARI) /* By definition: */
%o A246509 {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)}
%o A246509 for(n=0, 25, print1(a(n), ", "))
%o A246509 (PARI) /* By a binomial identity: */
%o A246509 {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)}
%o A246509 for(n=0, 25, print1(a(n), ", "))
%o A246509 (PARI) /* By a binomial identity: */
%o A246509 {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)}
%o A246509 for(n=0, 25, print1(a(n), ", "))
%o A246509 (PARI) /* By a binomial identity: */
%o A246509 {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)}
%o A246509 for(n=0, 25, print1(a(n), ", "))
%o A246509 (PARI) /* Formula for a(n): */
%o A246509 {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))}
%o A246509 for(n=0, 25, print1(a(n), ", "))
%Y A246509 Cf. A246510, A246423.
%K A246509 nonn
%O A246509 0,2
%A A246509 _Paul D. Hanna_, Aug 27 2014