OFFSET
1,2
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..500
A. O. L. Atkin, P. Bratley, I. G. McDonald, and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald, and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
Srivatsan Balakrishnan, Suresh Govindarajan, and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011, p.21.
N. J. A. Sloane, Transforms
FORMULA
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
MAPLE
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
MATHEMATICA
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 *)
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved