A248055 G.f.: Sum_{n>=0} x^n / (1-4*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k].

1, 5, 34, 265, 2219, 19490, 177119, 1651405, 15707416, 151791375, 1485814989, 14697965770, 146673432721, 1474490991635, 14915914368896, 151701887367585, 1550083118902041, 15903333300738320, 163749905809635219, 1691449817705302875, 17521670544878571584, 181972459046153912945

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

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) * x^k].

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) * x^j.

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 * 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 * 4^j.

a(n) ~ (11+3*sqrt(13))^(n+1) / (Pi * n * 2^(n+7/2)). - Vaclav Kotesovec, Oct 04 2014

Example

G.f.: A(x) = 1 + 5*x + 34*x^2 + 265*x^3 + 2219*x^4 + 19490*x^5 +...

Mathematica

Table[Sum[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 *)

Prog

(PARI) /* By definition: */
{a(n, p=4, q=1)=local(A=1); A=sum(m=0, n, x^m/(1-p*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * q^k * x^k) * sum(k=0, m, binomial(m, k)^2 * p^k * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n, 4, 1), ", "))
(PARI) /* By a binomial identity: */
{a(n, p=4, q=1)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * p^(m-k) * q^k * x^k) * sum(k=0, m, binomial(m, k)^2 * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n, 4, 1), ", "))
(PARI) /* By a binomial identity: */
{a(n, p=4, q=1)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * p^(k-j) * q^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n, 4, 1), ", "))
(PARI) /* By a binomial identity: */
{a(n, p=4, q=1)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * p^(m-k) * sum(j=0, k, binomial(k, j)^2 * q^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n, 4, 1), ", "))
(PARI) /* Formula for a(n): */
{a(n, p=4, q=1)=sum(k=0, n\2, sum(j=0, n-2*k, q^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * p^j))}
for(n=0, 25, print1(a(n, 4, 1), ", "))

Crossrefs

Cf. A246510, A248053, A246423, A246455, A246056, A246813.

Sequence in context: A002776 A333804 A081342 * A365218 A243659 A365183

Adjacent sequences: A248052 A248053 A248054 * A248056 A248057 A248058

Keyword

nonn

Author

Paul D. Hanna, Sep 30 2014

Status

approved