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%I M3859 N1581 #53 Nov 18 2022 08:55:44
%S 1,5,15,45,120,331,855,2214,5545,13741,33362,80091,189339,442799,
%T 1023192,2340904,5302061,11902618,26488454,58479965,128120214,
%U 278680698,602009786,1292027222,2755684669,5842618668,12317175320,25825429276,53865355154,111786084504,230867856903,474585792077,971209629993
%N Euler transform of A000292.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000335/b000335.txt">Table of n, a(n) for n=1..500</a>
%H A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="http://dx.doi.org/10.1017/S0305004100042171">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
%H A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="/A000219/a000219.pdf">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
%H Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, <a href="http://arxiv.org/abs/1105.6231">On the asymptotics of higher-dimensional partitions</a>, arXiv:1105.6231 [cond-mat.stat-mech], 2011, p. 20.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 - Zeta(3) / (8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8*Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2 / (90*Zeta(5)) + Zeta'(-3)/6 + (Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) - Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5)) - Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5))) * n^(2/5) + Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)). - _Vaclav Kotesovec_, Mar 12 2015
%p with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> binomial(n+2,3)): seq(a(n), n=1..26); # _Alois P. Heinz_, Sep 08 2008
%t max = 33; f[x_] := Exp[ Sum[ x^k/(1-x^k)^4/k, {k, 1, max}]]; Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x], 1](* _Jean-François Alcover_, Nov 21 2011, after _Joerg Arndt_ *)
%t nmax=50; Rest[CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)/6),{k,1,nmax}],{x,0,nmax}],x]] (* _Vaclav Kotesovec_, Mar 11 2015 *)
%t etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, #*p[#] &]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Binomial[#+2, 3]&]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Nov 24 2015, after _Alois P. Heinz_ *)
%o (PARI) a(n)=if(n<1, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^4/k, x*O(x^n))), n)) /* _Joerg Arndt_, Apr 16 2010 */
%o (PARI) N=66; x='x+O('x^66); gf=-1 + exp(sum(k=1, N, x^k/(1-x^k)^4/k)); Vec(gf) /* _Joerg Arndt_, Jul 06 2011 */
%o (Sage) # uses[EulerTransform from A166861] and prepends a(0) = 1.
%o a = EulerTransform(lambda n: n*(n+1)*(n+2)//6)
%o print([a(n) for n in range(33)]) # _Peter Luschny_, Nov 17 2022
%Y Cf. A000041, A000219, A000294, A000391, A000417, A000428, A255965.
%K nonn,nice
%O 1,2
%A _N. J. A. Sloane_
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