A249513 Expansion of -(4*x*sqrt(4*x^2+1)+8*x^2+1)/((2*x^2-1)*sqrt(4*x^2+1) +4*x^3+x).

1, 5, 13, 25, 51, 125, 295, 625, 1345, 3125, 7173, 15625, 34269, 78125, 177153, 390625, 864315, 1953125, 4401655, 9765625, 21706831, 48828125, 109676283, 244140625, 544031251, 1220703125, 2736797215, 6103515625, 13620096675, 30517578125, 68346531855

Offset

0,2

Links

Table of n, a(n) for n=0..30.

Formula

a(n) = sum(k = 0..n, 4^(n-k)*binomial((n+1)/2,n-k)).

a(n) ~ 5^((n+1)/2). - Vaclav Kotesovec, Oct 31 2014

a(n) = 5^((n+1)/2) if n is odd else a(n) = (4^n*binomial(n/2+1/2, n)* hypergeometric([1, -n], [-n/2+3/2], -1/4)). # Peter Luschny, Oct 31 2014

Conjecture D-finite with recurrence: -n*(3*n-4)*a(n) -(n-1)*(3*n-7)*a(n-1) +(3*n^2+8*n+24)*a(n-2) +(3*n^2+2*n+19)*a(n-3) +20*(3*n+2)*(n-3)*a(n-4) +20*(3*n-1)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 25 2020

Mathematica

CoefficientList[Series[-(4 x Sqrt[4 x^2 + 1] + 8 x^2 + 1)/((2 x^2 - 1) Sqrt[4 x^2 + 1] + 4 x^3 + x), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 31 2014 *)
Table[Sum[4^(n-k) Binomial[(n+1)/2, n-k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Jun 10 2023 *)

Prog

(Maxima)
a(n) := sum(4^(n-k)*binomial((n+1)/2, n-k), k, 0, n).
(Sage)
def a(n):
    if is_odd(n): return 5^(n//2+1)
    return (4^n*binomial(n/2+1/2, n)*hypergeometric([1, -n], [-n/2 +3/2], -1/4)).simplify_hypergeometric()
[a(n) for n in range(31)] # Peter Luschny, Oct 31 2014

Crossrefs

Cf. A000351.

Sequence in context: A147427 A147408 A241657 * A147090 A097117 A146140

Adjacent sequences: A249510 A249511 A249512 * A249514 A249515 A249516

Keyword

nonn

Author

Vladimir Kruchinin, Oct 31 2014

Extensions

More terms from Vincenzo Librandi, Oct 31 2014

Status

approved