Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #6 Oct 04 2014 05:34:47
%S 1,5,36,305,2821,27690,282699,2967285,31785786,345815975,3808549531,
%T 42360017130,474990254821,5362633500755,60897115958286,
%U 695012481567465,7966829676299139,91674042449673960,1058486539560201051,12258669983923625475,142359286920427682046,1657287004720545992505
%N 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].
%C 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.
%C 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.
%F 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].
%F 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.
%F 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.
%F 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.
%F 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
%e G.f.: A(x) = 1 + 5*x + 36*x^2 + 305*x^3 + 2821*x^4 + 27690*x^5 +...
%t 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 *)
%o (PARI) /* By definition: */
%o {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)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* By a binomial identity: */
%o {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)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* By a binomial identity: */
%o {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)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* By a binomial identity: */
%o {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)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* Formula for a(n): */
%o {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))}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A246509, A246056, A074938, A074939.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 27 2014