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 A002597 Number of partitions into one kind of 1's, two kinds of 2's, and three kinds of 3's. (Formerly M2533 N1000) 9
 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 (list; graph; refs; listen; history; text; internal format)
 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] 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: A082004 A112773 A070885 * A287554 A057855 A198685 Adjacent sequences:  A002594 A002595 A002596 * A002598 A002599 A002600 KEYWORD nonn AUTHOR EXTENSIONS More terms from Henry Bottomley, Sep 17 2001 Better name from Joerg Arndt, Oct 12 2012 STATUS approved

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