A002597 Number of partitions into one kind of 1's, two kinds of 2's, and three kinds of 3's.

1, 1, 3, 6, 9, 15, 25, 34, 51, 73, 97, 132, 178, 226, 294, 376, 466, 582, 722, 872, 1062, 1282, 1522, 1812, 2147, 2507, 2937, 3422, 3947, 4557, 5243, 5978, 6825, 7763, 8771, 9912, 11172, 12516, 14028, 15680, 17444, 19404, 21540, 23808, 26316, 29028, 31908

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• changed

Comments

Old name was: A generalized partition function.

References

• Gupta, Hansraj; A generalization of the partition function. Proc. Nat. Inst. Sci. India 17, (1951). 231-238.
• 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 = 0..1000
• H. Gupta, A generalization of the partition function, Proc. Nat. Inst. Sci. India 17, (1951). 231-238. [Annotated scanned copy]
• Index entries for linear recurrences with constant coefficients, signature (1, 2, 1, -4, -5, 3, 6, 3, -5, -4, 1, 2, 1, -1).

Crossrefs

Cf. A064349.

Programs

Maple

a:= proc(n) option remember;
     `if`(n=0, 1, add(add(d *`if`(d<4, d, 0),
      d=numtheory[divisors](j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 21 2012

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*If[d<4, d, 0], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)
LinearRecurrence[{1,2,1,-4,-5,3,6,3,-5,-4,1,2,1,-1},{1,1,3,6,9,15,25,34,51,73,97,132,178,226},50] (* Harvey P. Dale, Aug 28 2025 *)

PARI

a(n)=round((n\3+1)*((n\3+4)*[1,-1,0][1+n%3]/18-(n%3>1)/27)+(n+1)*(2*n^4+68*n^3+852*n^2+4748*n+10735+405*(-1)^n)/25920) \\ Tani Akinari, May 29 2014

Formula

G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)^3). - Henry Bottomley, Sep 17 2001

Euler transform of [1, 2, 3, 0, 0, 0, 0, 0, ...]. - Thomas Wieder, Mar 13 2005

a(n)=floor((160*(n+1)*(-1)^(floor(n/3+2/3)+n)+80*(n^2+15*n+24)*(-1)^(floor(n/3+1/3)+n)+80*(n+2)*(n+11)*(-1)^(floor(n/3)+n)+405*(n+1)*(-1)^n+(n+1)*(2*n^4+68*n^3+852*n^2+4748*n+10735))/25920+1/2). - Tani Akinari, Oct 12 2012

Extensions

More terms from Henry Bottomley, Sep 17 2001

Better name from Joerg Arndt, Oct 12 2012