cp's OEIS Frontend

A246812 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^k * x^k].

%I A246812 #5 Sep 03 2014 20:20:07
%S A246812 1,2,9,44,241,1374,8145,49512,306729,1927802,12256753,78661620,
%T A246812 508786129,3312561638,21688815729,142699137072,942873631497,
%U A246812 6253352120322,41611854129585,277723513754364,1858529465302329,12467403845702526,83817799189753785,564633483609422808,3810607016379076521
%N A246812 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^k * x^k].
%F A246812 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 3^j * x^j.
%F A246812 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.
%e A246812 G.f.: A(x) = 1 + 2*x + 9*x^2 + 44*x^3 + 241*x^4 + 1374*x^5 + 8145*x^6 +...
%e A246812 where the g.f. is given by the binomial series:
%e A246812 A(x) = 1/(1-x) + x/(1-x)^3*(1+x) * (1+3*x)
%e A246812 + x^2/(1-x)^5*(1 + 2^2*x + x^2) * (1 + 2^2*3*x + 9*x^2)
%e A246812 + x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3) * (1 + 3^2*3*x + 3^2*9*x^2 + 27*x^3)
%e A246812 + x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (1 + 4^2*3*x + 6^2*9*x^2 + 4^2*27*x^3 + 81*x^4)
%e A246812 + x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5) * (1 + 5^2*3*x + 10^2*9*x^2 + 10^2*27*x^3 + 5^2*81*x^4 + 243*x^5) +...
%e A246812 We can also express the g.f. by the binomial series identity:
%e A246812 A(x) = 1 + x*(1 + (1+3*x)) + x^2*(1 + 2^2*(1+3*x) + (1+2^2*3*x+9*x^2))
%e A246812 + x^3*(1 + 3^2*(1+3*x) + 3^2*(1+2^2*3*x+9*x^2) + (1+3^2*3*x+3^2*9*x^2+27*x^3))
%e A246812 + x^4*(1 + 4^2*(1+3*x) + 6^2*(1+2^2*3*x+9*x^2) + 4^2*(1+3^2*3*x+3^2*9*x^2+27*x^3) + (1+4^2*3*x+6^2*9*x^2+4^2*27*x^3+81*x^4))
%e A246812 + x^5*(1 + 5^2*(1+3*x) + 10^2*(1+2^2*3*x+9*x^2) + 10^2*(1+3^2*3*x+3^2*9*x^2+27*x^3) + 5^2*(1+4^2*3*x+6^2*9*x^2+4^2*27*x^3+81*x^4) + (1+5^2*3*x+10^2*9*x^2+10^2*27*x^3+5^2*81*x^4+243*x^5)) +...
%o A246812 (PARI) /* By definition: */
%o A246812 {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 * x^k) * sum(k=0, m, binomial(m, k)^2 * 3^k * x^k) +x*O(x^n)); polcoeff(A, n)}
%o A246812 for(n=0, 25, print1(a(n), ", "))
%o A246812 (PARI) /* By a binomial identity: */
%o A246812 {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^j * x^j)+x*O(x^n))), n)}
%o A246812 for(n=0, 25, print1(a(n), ", "))
%o A246812 (PARI) /* Formula for a(n): */
%o A246812 {a(n)=sum(k=0, n\2, 3^k * sum(j=0, n-2*k, binomial(n-k, k+j)^2 * binomial(k+j, j)^2))}
%o A246812 for(n=0, 25, print1(a(n), ", "))
%Y A246812 Cf. A246813 (dual), A243948, A246455, A246056, A246423, A246539, A245929, A227845, A245925.
%K A246812 nonn
%O A246812 0,2
%A A246812 _Paul D. Hanna_, Sep 03 2014