

A054766


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


5



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(n2)+3)*t[[1]]+(n1)^2*t[[2]]], {n, 2, 18}]; t (* Indranil Ghosh, Feb 25 2017 *)


CROSSREFS

Cf. A012244, A054765.
Sequence in context: A222508 A220841 A343425 * A252830 A301434 A232192
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



