A246653 G.f.: Sum_{n>=0} x^n / (1-x^2)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * x^(2*k)].

1, 1, 3, 9, 25, 87, 287, 993, 3519, 12525, 45369, 165519, 608569, 2253307, 8386575, 31370297, 117834837, 444258387, 1680516731, 6375706325, 24253227159, 92481509389, 353417696625, 1353280137975, 5191349266275, 19948136148837, 76771074483837, 295880290515411, 1141860205058433, 4412129801125275

Offset

0,3

Links

Table of n, a(n) for n=0..29.

Formula

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(n-k) * Sum_{j=0..k} C(k,j)^2 * x^j.

a(n) = Sum_{k=0..[n/2]} binomial(2*k, k) * binomial(n-k, k)^2.

Recurrence: n^2*(4*n-9)*a(n) = (8*n^3 - 26*n^2 + 18*n - 5)*a(n-1) + (28*n^3 - 119*n^2 + 162*n - 75)*a(n-2) + 2*(16*n^3 - 84*n^2 + 142*n - 75)*a(n-3) - 4*(2*n-5)^2*(4*n-5)*a(n-4). - Vaclav Kotesovec, Sep 02 2014

a(n) ~ 2^(2*n+1/2) / (Pi*n). - Vaclav Kotesovec, Sep 02 2014

Example

G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 25*x^4 + 87*x^5 + 287*x^6 + 993*x^7 +...
where the g.f. is given by the binomial series:
A(x) = 1/(1-x^2) + x/(1-x^2)^3 * (1 + x) * (1 + x^2)
+ x^2/(1-x^2)^5 * (1 + 2^2*x + x^2) * (1 + 2^2*x^2 + x^4)
+ x^3/(1-x^2)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (1 + 3^2*x^2 + 3^2*x^4 + x^6)
+ x^4/(1-x^2)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (1 + 4^2*x^2 + 6^2*x^4 + 4^2*x^6 + x^8)
+ x^5/(1-x^2)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5) * (1 + 5^2*x^2 + 10^2*x^4 + 10^2*x^6 + 5^2*x^8 + x^10) +...

Mathematica

Table[Sum[Binomial[2*k, k] * Binomial[n-k, k]^2, {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 02 2014 *)

Prog

(PARI) /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x^2)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^k) * sum(k=0, m, binomial(m, k)^2 * x^(2*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, 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 * x^(m-k) * sum(j=0, k, binomial(k, j)^2 * x^j )+x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From a formula for a(n): */
{a(n)=sum(k=0, n\2, binomial(2*k, k)*binomial(n-k, k)^2)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A246563.

Sequence in context: A101786 A354647 A217995 * A192371 A363467 A192465

Adjacent sequences: A246650 A246651 A246652 * A246654 A246655 A246656

Keyword

nonn

Author

Paul D. Hanna, Aug 31 2014

Status

approved