A002597 Number of partitions into one kind of 1's, two kinds of 2's, and three kinds of 3's. (Formerly M2533 N1000)

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

0,3

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A064349.

Sequence in context: A363615 A357989 A070885 * A360624 A287554 A308777

Adjacent sequences: A002594 A002595 A002596 * A002598 A002599 A002600

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Henry Bottomley, Sep 17 2001

Better name from Joerg Arndt, Oct 12 2012

Status

approved