A000391 Euler transform of A000332. (Formerly M4144 N1721)

1, 6, 21, 71, 216, 672, 1982, 5817, 16582, 46633, 128704, 350665, 941715, 2499640, 6557378, 17024095, 43756166, 111433472, 281303882, 704320180, 1749727370, 4314842893, 10565857064, 25700414815, 62115621317, 149214574760, 356354881511, 846292135184

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

Cf. A000041, A000219, A000294, A000335, A000417, A000428, A255965.

Sequence in context: A101904 A022814 A000390 * A360090 A107660 A200665

Adjacent sequences: A000388 A000389 A000390 * A000392 A000393 A000394

Keyword

nonn,changed

Author

N. J. A. Sloane

Status

approved