A322245 a(n) = A322244(n)^2, the square of the central coefficient in (1 + 3*x + 16x^2)^n.

1, 9, 1681, 99225, 11189025, 1019077929, 108167974321, 11137724806329, 1197175570050625, 129043286745662025, 14139849178569444561, 1560056939689534791129, 173632800433761049827681, 19441545948416137605093225, 2189471746316530595782640625, 247739375533243457859858956025, 28151322919072607132343448452225, 3210745922813847247161044673062025

Offset

0,2

Comments

The g.f. of A322244 is 1/sqrt(1 - 6*x - 55*x^2).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..481

Formula

G.f.: 1 / AGM(1 + 5*11*x, sqrt((1 - 5^2*x)*(1 - 11^2*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.

G.f.: 1 / AGM((1-5*x)*(1-11*x), (1+5*x)*(1+11*x)) = Sum_{n>=0} a(n)*x^(2*n).

a(n) = A322244(n)^2, where A322244(n) = a(n) = Sum_{k=0..n} 11^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).

a(n) ~ 11^(2*n + 1) / (16*Pi*n). - Vaclav Kotesovec, Dec 13 2018

Example

G.f.: A(x) = 1 + 9*x + 1681*x^2 + 99225*x^3 + 11189025*x^4 + 1019077929*x^5 + 108167974321*x^6 + 11137724806329*x^7 + 1197175570050625*x^8 + ...
that is,
A(x) = 1 + 3^2*x + 41^2*x^2 + 315^2*x^3 + 3345^2*x^4 + 31923^2*x^5 + 328889^2*x^6 + 3337323^2*x^7 + 34600225^2*x^8 + ... + A322244(n)^2*x^n + ...

Mathematica

a[n_] := Sum[11^(n-k) * (-4)^k * Binomial[n, k] * Binomial[2k, k], {k, 0, n}]^2; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)

Prog

(PARI) /* a(n) = A322244(n)^2 - g.f. */
{a(n)=polcoeff(1/sqrt(1 - 6*x - 55*x^2 +x*O(x^n)), n)^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* a(n) = A322244(n)^2 - binomial sum */
{a(n) = sum(k=0, n, 11^(n-k)*(-4)^k*binomial(n, k)*binomial(2*k, k))^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Using AGM: */
{a(n)=polcoeff( 1 / agm(1 + 5*11*x, sqrt((1 - 5^2*x)*(1 - 11^2*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A322244.

Sequence in context: A197206 A197804 A047944 * A251700 A246115 A068182

Adjacent sequences: A322242 A322243 A322244 * A322246 A322247 A322248

Keyword

nonn

Author

Paul D. Hanna, Dec 09 2018

Status

approved