A028353 Coefficient of x^(2*n+1) in arctanh(x)/sqrt(1-x^2), multiplied by (2*n+1)!.

1, 5, 89, 3429, 230481, 23941125, 3555578025, 715154761125, 187188449198625, 61836509511685125, 25163273966324405625, 12368068140988819153125, 7224011282550809645600625

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

Cf. A060338.

Cf. A060524.

Programs

Mathematica

Table[n!*SeriesCoefficient[ArcTanh[x]/Sqrt[1-x^2],{x,0,n}],{n,1,41,2}] (* Vaclav Kotesovec, Oct 24 2013 *)

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