A000417 Euler transform of A000389. (Formerly M4392 N1849)

1, 7, 28, 105, 357, 1232, 4067, 13301, 42357, 132845, 409262, 1243767, 3727360, 11036649, 32300795, 93538278, 268164868, 761656685, 2144259516, 5986658951, 16583102077, 45593269265, 124464561544, 337479729179, 909156910290, 2434121462871, 6478440788169

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..1000

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]

N. J. A. Sloane, Transforms

Formula

a(n) ~ (3*Zeta(7))^(31103/423360) / (2^(180577/423360) * sqrt(7*Pi) * n^(242783/423360)) * exp(Zeta'(-1)/5 - 5*Zeta(3)/(48*Pi^2) + Zeta(5)/(16*Pi^4) - Pi^36/(1162964338810860915 * Zeta(7)^5) + Pi^24 * Zeta(5) / (413420708484 * Zeta(7)^4) - Pi^22 / (137806902828 * Zeta(7)^3) - Pi^12 * Zeta(5)^2 / (551124 * Zeta(7)^3) + Pi^12 * Zeta(3) / (11252115 * Zeta(7)^2) + Pi^10 * Zeta(5) / (122472 * Zeta(7)^2) + 49*Zeta(5)^3 / (216 * Zeta(7)^2) - Pi^8 / (108864 * Zeta(7)) - Zeta(3) * Zeta(5) / (15*Zeta(7)) + Zeta'(-5)/120 + 7*Zeta'(-3)/24 + (22 * 2^(6/7) * Pi^30 / (46901442470561469 * 3^(1/7) * Zeta(7)^(29/7)) - 10 * 2^(6/7) * Pi^18 * Zeta(5) / (8931928887 * 3^(1/7) * Zeta(7)^(22/7)) + Pi^16 / (141776649 * 6^(1/7) * Zeta(7)^(15/7)) + 2^(6/7) * Pi^6 * Zeta(5)^2 / (1701 * 3^(1/7) * Zeta(7)^(15/7)) - 2^(6/7) * Pi^6 * Zeta(3) / (19845 * 3^(1/7) * Zeta(7)^(8/7)) - Pi^4 * Zeta(5) / (216 * 6^(1/7) * Zeta(7)^(8/7))) * n^(1/7) + (-2 * 2^(5/7) * Pi^24 / (3938980639167 * 3^(2/7) * Zeta(7)^(23/7)) + Pi^12 * Zeta(5) / (500094 * 6^(2/7) * Zeta(7)^(16/7)) - Pi^10 / (142884 * 6^(2/7) * Zeta(7)^(9/7)) - 7*Zeta(5)^2 / (12 * 6^(2/7) * Zeta(7)^(9/7)) + Zeta(3)/(5 * (6*Zeta(7))^(2/7))) * n^(2/7) + (5 * 2^(4/7) * Pi^18 / (8931928887 * 3^(3/7) * Zeta(7)^(17/7)) - Pi^6 * Zeta(5) / (567 * 6^(3/7) * Zeta(7)^(10/7)) + Pi^4 / (108 * (6*Zeta(7))^(3/7))) * n^(3/7) + (-Pi^12 / (750141 * 6^(4/7) * Zeta(7)^(11/7)) + 7*Zeta(5) / (4 * (6 * Zeta(7))^(4/7))) * n^(4/7) + 2^(2/7) * Pi^6 / (945 * (3*Zeta(7))^(5/7)) * n^(5/7) + 7*Zeta(7)^(1/7) / 6^(6/7) * n^(6/7)). - 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+4, 5)): seq(a(n), n=1..30); # Alois P. Heinz, Sep 08 2008

Mathematica

nn = 100; b = Table[Binomial[n, 5], {n, 5, nn + 5}]; Rest[CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x]] (* T. D. Noe, Jun 20 2012 *)

Prog

(PARI) a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^6/k, x*O(x^n))), n)) /* Joerg Arndt, Apr 16 2010 */

Crossrefs

Cf. A000041, A000219, A000294, A000335, A000391, A000428, A255965.

Sequence in context: A331197 A024207 A000416 * A200762 A243150 A344206

Adjacent sequences: A000414 A000415 A000416 * A000418 A000419 A000420

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Nov 14 2010

Status

approved