A246813 G.f.: Sum_{n>=0} x^n / (1-3*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, 4, 23, 152, 1085, 8156, 63579, 509136, 4161649, 34566580, 290798551, 2471871784, 21191824645, 182984610220, 1589620392835, 13881368684128, 121767703088377, 1072382299895428, 9477296423786207, 84017470425706040, 746903374745524629, 6656552616997851036, 59459592374756968323

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

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 * x^j.

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

a(n) ~ sqrt(12 + 23/sqrt(3) + 2*sqrt(80 + 46*sqrt(3))) * (3 + sqrt(3) + sqrt(11 + 6*sqrt(3)))^n / (4*Pi*n). - Vaclav Kotesovec, Oct 04 2014

Example

G.f.: A(x) = 1 + 4*x + 23*x^2 + 152*x^3 + 1085*x^4 + 8156*x^5 +...
where the g.f. is given by the binomial series:
A(x) = 1/(1-3*x) + x/(1-3*x)^3 * (1+x) * (1+3*x)
+ x^2/(1-3*x)^5 * (1 + 2^2*x + x^2) * (1 + 2^2*3*x + 9*x^2)
+ x^3/(1-3*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/(1-3*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/(1-3*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:
A(x) = 1 + x*(1 + (3+x)) + x^2*(1 + 2^2*(3+x) + (9+2^2*3*x+x^2))
+ x^3*(1 + 3^2*(3+x) + 3^2*(9+2^2*3*x+x^2) + (27+3^2*9*x+3^2*3*x^2+x^3))
+ x^4*(1 + 4^2*(3+x) + 6^2*(9+2^2*3*x+x^2) + 4^2*(27+3^2*9*x+3^2*3*x^2+x^3) + (81+4^2*27*x+6^2*9*x^2+4^2*3*x^3+x^4))
+ x^5*(1 + 5^2*(3+x) + 10^2*(9+2^2*3*x+x^2) + 10^2*(27+3^2*9*x+3^2*3*x^2+x^3) + 5^2*(81+4^2*27*x+6^2*9*x^2+4^2*3*x^3+x^4) + (243+5^2*81*x+10^2*27*x^2+10^2*9*x^3+5^2*3*x^4+x^5)) +...

Mathematica

Table[Sum[Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 3^j, {j, 0, n-2*k}], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 04 2014 *)

Prog

(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 * 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 * 3^(m-k) * sum(j=0, k, binomial(k, j)^2 * 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, binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 3^j))}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A246812 (dual), A246455, A243948, A246056, A246423, A246539, A245929, A227845, A245925.

Sequence in context: A350480 A193113 A192730 * A369213 A055723 A364478

Adjacent sequences: A246810 A246811 A246812 * A246814 A246815 A246816

Keyword

nonn

Author

Paul D. Hanna, Sep 03 2014

Status

approved