OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..962
FORMULA
a(n) = Sum_{k=0..n} 11^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
a(n) = Sum_{k=0..n} (-5)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).
a(n) equals the (central) coefficient of x^n in (1 + 3*x + 16*x^2)^n.
a(n) ~ 11^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 13 2018
D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +55*(-n+1)*a(n-2)=0. - R. J. Mathar, Jan 16 2020
EXAMPLE
G.f.: A(x) = 1 + 3*x + 41*x^2 + 315*x^3 + 3345*x^4 + 31923*x^5 + 328889*x^6 + 3337323*x^7 + 34600225*x^8 + 359225955*x^9 + 3760299081*x^10 + ...
such that A(x)^2 = 1/(1 - 6*x - 55*x^2).
RELATED SERIES.
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 3*x + 25*x^2 + 171*x^3 + 1457*x^4 + 12243*x^5 + 109769*x^6 + 997755*x^7 + 9314657*x^8 + 88177059*x^9 + 847159161*x^10 + ...
MATHEMATICA
a[n_] := Sum[(-5)^(n-k) * 4^k * Binomial[n, k] * Binomial[2k, k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)
CoefficientList[Series[1/Sqrt[1-6x-55x^2], {x, 0, 40}], x] (* Harvey P. Dale, Aug 13 2024 *)
PROG
(PARI) /* Using generating function: */
{a(n) = polcoeff( 1/sqrt(1 - 6*x - 55*x^2 +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Using binomial formula: */
{a(n) = sum(k=0, n, (-5)^(n-k)*4^k*binomial(n, k)*binomial(2*k, k))}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Using binomial formula: */
{a(n) = sum(k=0, n, 11^(n-k)*(-4)^k*binomial(n, k)*binomial(2*k, k))}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* a(n) is central coefficient in (1 + 3*x + 4*x^2)^n */
{a(n) = polcoeff( (1 + 3*x + 16*x^2 +x*O(x^n))^n, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Dec 09 2018
STATUS
approved