A054766 - OEIS

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

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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