A036242 Numerator of fraction equal to the continued fraction [0,2,4,...2n].

1, 4, 25, 204, 2065, 24984, 351841, 5654440, 102131761, 2048289660, 45164504281, 1085996392404, 28281070706785, 792955976182384, 23816960356178305, 762935687373888144, 25963630331068375201, 935453627605835395380, 35573201479352813399641, 1423863512801718371381020

Offset: 1

Author

Jeff Burch

Keywords

• frac
• nonn
• easy

Crossrefs

Cf. A036243 (denominator), A058798.

Programs

Mathematica

a[n_] := FromContinuedFraction[Range[0, 2n, 2]] // Numerator; Array[a,20] (* Jean-François Alcover, Jun 03 2019 *)

Sage

def A036242(n):
    if n == 1: return 1
    return 2^(n-1)*factorial(n)*hypergeometric([1/2 - n/2, 1 - n/2], [2, 1-n, -n], 1)
[round(A036242(n).n(100)) for n in (1..20)] # Peter Luschny, Sep 14 2014

Formula

Recurrence equation: a(n+1) = (2*n+2)*a(n) + a(n-1) with a(0) = 1 and a(1) = 1.

a(n) = Sum_{k = 0..floor((n-1)/2)} 2^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k,k+1). Cf. A058798. - Peter Bala, Aug 01 2013

a(n) = 2^(n-1)*n!*hypergeometric([(1-n)/2, 1-n/2],[2, 1-n, -n], 1) for n>=2. - Peter Luschny, Sep 14 2014