|
|
A337397
|
|
Expansion of sqrt(2 / ( (1+64*x^2) * (1-8*x+sqrt(1+64*x^2)) )).
|
|
3
|
|
|
1, 2, -34, -92, 1654, 4828, -88724, -268088, 4984486, 15361708, -287691196, -898052872, 16901635516, 53234639768, -1005474931816, -3187958034544, 60375963282182, 192405594166988, -3651655920615596, -11684176213422568, 222132094724096852, 713091439789994824, -13575872676384218776
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).
a(0) = 1, a(1) = 2 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * 2 * a(n-1) - 64 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020
|
|
MATHEMATICA
|
a[n_] := Sum[(-4)^(n - k) * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 26 2020 *)
CoefficientList[Series[Sqrt[2/((1+64x^2)(1-8x+Sqrt[1+64x^2]))], {x, 0, 30}], x] (* Harvey P. Dale, Jul 24 2021 *)
|
|
PROG
|
(PARI) N=40; x='x+O('x^N); Vec(sqrt(2/((1+64*x^2)*(1-8*x+sqrt(1+64*x^2)))))
(PARI) {a(n) = sum(k=0, n, (-4)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|