A322249 a(n) = A322248(n)^2, the square of the central coefficient in (1 + 5*x + 16*x^2)^n.

1, 25, 3249, 366025, 48455521, 6646325625, 947789867025, 138422872355625, 20598606105401025, 3109408600719825625, 474780862425986767729, 73175222677868396505225, 11366022325041154078402081, 1777059915791491092441607225, 279404859303904515406536763089, 44144451113336819125597110875625, 7004264626817806908496520658161025, 1115512654966236899680358546064905625

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..450

Formula

G.f.: 1 / AGM(1 + 3*13*x, sqrt((1 - 3^2*x)*(1 - 13^2*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.

G.f.: 1 / AGM((1-3*x)*(1-13*x), (1+3*x)*(1+13*x)) = Sum_{n>=0} a(n)*x^(2*n).

a(n) = A322248(n)^2, where A322248(n) = a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).

a(n) = A322248(n)^2, where A322248(n) = a(n) = Sum_{k=0..n} 13^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).

a(n) ~ 13^(2*n + 1) / (16*Pi*n). - Vaclav Kotesovec, Dec 10 2018

Example

G.f.: A(x) = 1 + 25*x + 3249*x^2 + 366025*x^3 + 48455521*x^4 + 6646325625*x^5 + 947789867025*x^6 + 138422872355625*x^7 + 20598606105401025*x^8 + ...
such that
A(x) = 1 + 5^2*x + 57^2*x^2 + 605^2*x^3 + 6961^2*x^4 + 81525^2*x^5 + 973545^2*x^6 + 11765325^2*x^7 + 143522145^2*x^8 + ... + A322248(n)^2*x^n + ...

Prog

(PARI) /* a(n) = A322248(n)^2 - g.f. */
{a(n)=polcoeff(1/sqrt((1 + 3*x)*(1 - 13*x) +x*O(x^n)), n)^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* a(n) = A322248(n)^2 - g.f. */
{a(n) = polcoeff( (1 + 5*x + 16*x^2 +x*O(x^n))^n, n)^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* a(n) = A322248(n)^2 - binomial sum */
{a(n) = sum(k=0, n, (-3)^(n-k)*4^k*binomial(n, k)*binomial(2*k, k))^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Using binomial formula: */
{a(n) = sum(k=0, n, 13^(n-k)*(-4)^k*binomial(n, k)*binomial(2*k, k))^2}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Using AGM: */
{a(n)=polcoeff( 1 / agm(1 + 3*13*x, sqrt((1 - 3^2*x)*(1 - 13^2*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A246467, A246906, A246923, A322248.

Sequence in context: A018861 A178188 A203186 * A373878 A185826 A178026

Adjacent sequences: A322246 A322247 A322248 * A322250 A322251 A322252

Keyword

nonn

Author

Paul D. Hanna, Dec 10 2018

Status

approved