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%I M4144 N1721 #42 Nov 28 2023 14:27:44
%S 1,6,21,71,216,672,1982,5817,16582,46633,128704,350665,941715,2499640,
%T 6557378,17024095,43756166,111433472,281303882,704320180,1749727370,
%U 4314842893,10565857064,25700414815,62115621317,149214574760,356354881511,846292135184
%N Euler transform of A000332.
%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="/A000391/b000391.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.21.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) ~ Pi^(3/160) / (2 * 3^(243/320) * 7^(83/960) * n^(563/960)) * exp(Zeta'(-1)/4 - 143 * Zeta(3) / (240 * Pi^2) + 53461 * Zeta(5) / (3200 * Pi^4) + 107163 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) - 24754653 * Zeta(5)^3 / (10*Pi^14) + 413420708484 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/4 + (-847 * 7^(1/6) * Pi / (19200 * sqrt(3)) - 189 * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5) / (2*Pi^7) + 305613 * sqrt(3) * 7^(1/6) * Zeta(5)^2 / (80*Pi^9) - 614365479 * sqrt(3) * 7^(1/6) * Zeta(5)^4 / (4*Pi^19)) * n^(1/6) + (3 * 7^(1/3) * Zeta(3) / (4*Pi^2) - 693 * 7^(1/3) * Zeta(5) / (40*Pi^4) + 857304 * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (11 * sqrt(7/3) * Pi / 120 - 1701 * sqrt(21) * Zeta(5)^2 / Pi^9) * sqrt(n) + 27 * 7^(2/3) * Zeta(5) / (2*Pi^4) * n^(2/3) + 2*sqrt(3)*Pi / (5*7^(1/6)) * n^(5/6)). - _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+3,4)): seq(a(n), n=1..30); # _Alois P. Heinz_, Sep 08 2008
%t nn = 50; b = Table[Binomial[n, 4], {n, 4, nn + 4}]; Rest[CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x]] (* _T. D. Noe_, Jun 21 2012 *)
%t nmax=50; Rest[CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)*(k+3)/24),{k,1,nmax}],{x,0,nmax}],x]] (* _Vaclav Kotesovec_, Mar 11 2015 *)
%o (PARI) a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^5/k, x*O(x^n))), n)) /* _Joerg Arndt_, Apr 16 2010 */
%Y Cf. A000041, A000219, A000294, A000335, A000417, A000428, A255965.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
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