cp's OEIS Frontend

A061256 Euler transform of sigma(n), cf. A000203.

%I A061256 #71 Sep 30 2024 17:38:50
%S A061256 1,1,4,8,21,39,92,170,360,667,1316,2393,4541,8100,14824,26071,46422,
%T A061256 80314,139978,238641,408201,686799,1156062,1920992,3189144,5238848,
%U A061256 8589850,13963467,22641585,36447544,58507590,93334008,148449417,234829969,370345918
%N A061256 Euler transform of sigma(n), cf. A000203.
%C A061256 This is also the number of ordered triples of permutations f, g, h in Symm(n) which all commute, divided by n!. This was conjectured by _Franklin T. Adams-Watters_, Jan 16 2006, and proved by J. R. Britnell in 2012.
%C A061256 According to a message on a blog page by "Allan" (see Secret Blogging Seminar link) it appears that a(n) = number of conjugacy classes of commutative ordered pairs in Symm(n).
%C A061256 John McKay (email to _N. J. A. Sloane_, Apr 23 2013) observes that A061256 and A006908 coincide for a surprising number of terms, and asks for an explanation. - _N. J. A. Sloane_, May 19 2013
%H A061256 Seiichi Manyama, <a href="/A061256/b061256.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A061256 Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://doi.org/10.1007/s44007-024-00134-w">A unified treatment of families of partition functions</a>, La Matematica (2024). Preprint available as <a href="https://arxiv.org/abs/2303.02240">arXiv:2303.02240</a> [math.CO], 2023.
%H A061256 J. R. Britnell, <a href="http://arXiv.org/abs/1203.5079">A formal identity involving commuting triples of permutations</a>, arXiv:1203.5079 [math.CO], 2012.
%H A061256 J. R. Britnell, <a href="http://wwwf.imperial.ac.uk/~jbritnel/CommTriples.pdf">A formal identity involving commuting triples of permutations</a>, Preprint 2012. - _N. J. A. Sloane_, Jun 13 2012
%H A061256 J. R. Britnell, <a href="https://doi.org/10.1016/j.jcta.2013.01.009">A formal identity involving commuting triples of permutations</a>, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013.
%H A061256 E. Marberg, <a href="http://arxiv.org/abs/1202.1311">How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system</a>, arXiv preprint arXiv:1202.1311 [math.RT], 2012. - _N. J. A. Sloane_, Jun 10 2012
%H A061256 Secret Blogging Seminar, <a href="https://sbseminar.wordpress.com/2010/10/06/a-peculiar-numerical-coincidence/">A peculiar numerical coincidence</a>.
%H A061256 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A061256 Tad White, <a href="http://arxiv.org/abs/1304.2830">Counting Free Abelian Actions</a>, arXiv:1304.2830 [math.CO], 2013.
%F A061256 a(n) = A072169(n) / n!.
%F A061256 G.f.: Product_{k=1..infinity} (1 - x^k)^(-sigma(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*sigma(d), cf. A001001.
%F A061256 G.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x^n)^2 /n ). [_Paul D. Hanna_, Mar 28 2009]
%F A061256 G.f.: exp( Sum_{n>=1} sigma_2(n)*x^n/(1-x^n)/n ). [_Vladeta Jovovic_, Mar 28 2009]
%F A061256 G.f.: prod(n>=1, E(x^n)^n ) where E(x) = prod(k>=1, 1-x^k). [_Joerg Arndt_, Apr 12 2013]
%F A061256 a(n) ~ exp((3*Pi)^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2 - Pi^(4/3) * n^(1/3) / (4 * 3^(2/3) * Zeta(3)^(1/3)) - 1/24 - Pi^2/(288*Zeta(3))) * A^(1/2) * Zeta(3)^(11/72) / (2^(11/24) * 3^(47/72) * Pi^(11/72) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Mar 23 2018
%e A061256 1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 92*x^6 + 170*x^7 + 360*x^8 + ...
%p A061256 with(numtheory):
%p A061256 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A061256       d*sigma(d), d=divisors(j)) *a(n-j), j=1..n)/n)
%p A061256     end:
%p A061256 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jun 08 2017
%t A061256 nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* _T. D. Noe_, Jun 18 2012 *)
%t A061256 nmax = 40; CoefficientList[Series[Product[1/QPochhammer[x^k]^k, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 29 2015 *)
%o A061256 (PARI) N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^j)^j); Vec(gf) /* _Joerg Arndt_, May 03 2008 */
%o A061256 (PARI) {a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^m+x*O(x^n))^2/m)),n))} /* _Paul D. Hanna_, Mar 28 2009 */
%Y A061256 Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), this sequence (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).
%Y A061256 Cf. A000203, A001001, A001970, A053529, A061255, A061257, A006908, A192065.
%K A061256 easy,nonn
%O A061256 0,3
%A A061256 _Vladeta Jovovic_, Apr 21 2001
%E A061256 Entry revised by _N. J. A. Sloane_, Jun 13 2012