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].

1, 2, 9, 44, 241, 1374, 8145, 49512, 306729, 1927802, 12256753, 78661620, 508786129, 3312561638, 21688815729, 142699137072, 942873631497, 6253352120322, 41611854129585, 277723513754364, 1858529465302329, 12467403845702526, 83817799189753785, 564633483609422808, 3810607016379076521

Offset

0,2

Links

Table of n, a(n) for n=0..24.

Formula

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.

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.

Example

G.f.: A(x) = 1 + 2*x + 9*x^2 + 44*x^3 + 241*x^4 + 1374*x^5 + 8145*x^6 +...
where the g.f. is given by the binomial series:
A(x) = 1/(1-x) + x/(1-x)^3*(1+x) * (1+3*x)
+ x^2/(1-x)^5*(1 + 2^2*x + x^2) * (1 + 2^2*3*x + 9*x^2)
+ 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)
+ 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)
+ 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) +...
We can also express the g.f. by the binomial series identity:
A(x) = 1 + x*(1 + (1+3*x)) + x^2*(1 + 2^2*(1+3*x) + (1+2^2*3*x+9*x^2))
+ 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))
+ 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))
+ 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)) +...

Prog

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

Crossrefs

Cf. A246813 (dual), A243948, A246455, A246056, A246423, A246539, A245929, A227845, A245925.

Sequence in context: A229189 A365129 A371576 * A365123 A108308 A119855

Adjacent sequences: A246809 A246810 A246811 * A246813 A246814 A246815

Keyword

nonn

Author

Paul D. Hanna, Sep 03 2014

Status

approved