A054766 a(n+2) = (2*n + 3)*a(n+1) + (n + 1)^2*a(n), a(0) = 1, a(1) = 0.

1, 0, 1, 5, 44, 476, 6336, 99504, 1803024, 37019664, 849418560, 21539756160, 598194037440, 18056575823040, 588622339549440, 20609136708249600, 771323264354361600, 30729606721005830400, 1298448658633614566400

Offset

0,4

Comments

Numerators of the convergents of the generalized continued fraction expansion 4/Pi - 1 = [0; 1/3, 4/5, 9/7,..., n^2/(2*n + 1),...] = 1/(3 + 4/(5 + 9/(7 + ...))). The first 4 convergents are 1/3, 5/19, 44/160 and 476/1744.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..392

K. S. Brown, Integer Sequences Related To Pi

Formula

a(n) ~ (1 - Pi/4) * (1 + sqrt(2))^(n + 1/2) * n^n / (2^(1/4) * exp(n)). - Vaclav Kotesovec, Feb 18 2017

Mathematica

RecurrenceTable[{a[n+2] == (2*n+3)*a[n+1] + (n+1)^2*a[n], a[0] == 1, a[1] == 0}, a, {n, 0, 25}] (* Vaclav Kotesovec, Feb 18 2017 *)
t={1, 0}; Do[AppendTo[t, (2(n-2)+3)*t[[-1]]+(n-1)^2*t[[-2]]], {n, 2, 18}]; t (* Indranil Ghosh, Feb 25 2017 *)

Crossrefs

Cf. A012244, A054765.

Sequence in context: A220841 A366924 A343425 * A369216 A252830 A301434

Adjacent sequences: A054763 A054764 A054765 * A054767 A054768 A054769

Keyword

nonn,easy,frac

Author

N. J. A. Sloane, May 26 2000

Extensions

More terms from James A. Sellers, May 27 2000

Definition expanded by Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

Keyword frac added by Michel Marcus, Feb 25 2017

Status

approved