A002134 Generalized divisor function. Number of partitions of n with exactly three part sizes. (Formerly M1367 N0530)

1, 2, 5, 10, 15, 25, 37, 52, 67, 97, 117, 154, 184, 235, 277, 338, 385, 469, 531, 630, 698, 810, 910, 1038, 1144, 1295, 1425, 1577, 1741, 1938, 2089, 2301, 2505, 2700, 2970, 3189, 3444, 3703, 4004, 4242, 4617, 4882, 5244, 5558, 5999, 6221, 6755, 7050, 7576

Offset

6,2

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

Alois P. Heinz, Table of n, a(n) for n = 6..10000

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Formula

G.f.: Sum_{i>=1} Sum_{j=1..i-1} Sum_{k=1..j-1} x^(i+j+k)/((1-x^i)*(1-x^j)* (1-x^k)). - Geoffrey Critzer, Sep 13 2012

Example

a(8) = 5 because we have 5+2+1, 4+3+1, 4+2+1+1, 3+2+2+1, 3+2+1+1+1.

Maple

# Using function P from A365676:
A002134 := n -> P(n, 3, n): seq(A002134(n), n = 6..54); # Peter Luschny, Sep 15 2023

Mathematica

nn=40; sss=Sum[Sum[Sum[x^(i+j+k)/(1-x^i)/(1-x^j)/(1-x^k), {k, 1, j-1}], {j, 1, i-1}], {i, 1, nn}]; Drop[CoefficientList[Series[sss, {x, 0, nn}], x], 6]  (* Geoffrey Critzer, Sep 13 2012 *)

Crossrefs

A diagonal of A060177.

Column k=3 of A116608. - Alois P. Heinz, Nov 07 2012

Sequence in context: A099738 A064513 A117582 * A243971 A062472 A135061

Adjacent sequences: A002131 A002132 A002133 * A002135 A002136 A002137

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Better description and more terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Nov 02 2003

Status

approved