OFFSET
0,2
COMMENTS
Number of degree-(2*n+1) permutations with exactly one odd cycle. - Vladeta Jovovic, Aug 13 2004
a(n)=sum over all multinomials M2(2*n+1,k), k from {1..p(2*n+1)} restricted to partitions with exactly one odd and possibly even parts. p(2*n+1)= A000041(2*n+1) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n+1,k). - Wolfdieter Lang, Aug 07 2007.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..220
Zhi-Hong Sun, Congruences for the Apéry numbers modulo p^3, arXiv:2409.06544 [math.NT], 2024. See t(n).
FORMULA
D-finite with recurrence: a(n) = (8*n^2 - 4*n + 1)*a(n-1) - 4*(n-1)^2*(2*n-1)^2*a(n-2). - Vaclav Kotesovec, Oct 24 2013
a(n) ~ (2*n)^(2*n+1)*log(n)/exp(2*n) * (1 + (gamma + 4*log(2)) / log(n)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 24 2013
EXAMPLE
arctanh(x)/sqrt(1-x^2) = x + 5/6*x^3 + 89/120*x^5 + 381/560*x^7 + ...
Multinomial representation for a(2): partitions of 2*2+1=5 with one odd part: (5) with position k=1, (1,4) with k=2, (2,3) with k=3, (1,2^2) with k=5; M2(5,1)= 24, M2(5,2)= 30, M2(5,3)= 20, M2(5,5)= 15, adding up to a(2)=89.
MATHEMATICA
Table[n!*SeriesCoefficient[ArcTanh[x]/Sqrt[1-x^2], {x, 0, n}], {n, 1, 41, 2}] (* Vaclav Kotesovec, Oct 24 2013 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Joe Keane (jgk(AT)jgk.org)
STATUS
approved