cp's OEIS Frontend

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

%I A028353 #28 Sep 11 2024 10:07:34
%S A028353 1,5,89,3429,230481,23941125,3555578025,715154761125,187188449198625,
%T A028353 61836509511685125,25163273966324405625,12368068140988819153125,
%U A028353 7224011282550809645600625
%N A028353 Coefficient of x^(2*n+1) in arctanh(x)/sqrt(1-x^2), multiplied by (2*n+1)!.
%C A028353 Number of degree-(2*n+1) permutations with exactly one odd cycle. - _Vladeta Jovovic_, Aug 13 2004
%C A028353 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.
%H A028353 G. C. Greubel, <a href="/A028353/b028353.txt">Table of n, a(n) for n = 0..220</a>
%H A028353 Zhi-Hong Sun, <a href="https://arxiv.org/abs/2409.06544">Congruences for the Apéry numbers modulo p^3</a>, arXiv:2409.06544 [math.NT], 2024. See t(n).
%F A028353 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
%F A028353 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
%e A028353 arctanh(x)/sqrt(1-x^2) = x + 5/6*x^3 + 89/120*x^5 + 381/560*x^7 + ...
%e A028353 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.
%t A028353 Table[n!*SeriesCoefficient[ArcTanh[x]/Sqrt[1-x^2],{x,0,n}],{n,1,41,2}] (* _Vaclav Kotesovec_, Oct 24 2013 *)
%Y A028353 Cf. A060338.
%Y A028353 Cf. A060524.
%K A028353 nonn,easy
%O A028353 0,2
%A A028353 Joe Keane (jgk(AT)jgk.org)