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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.

%I A002597 M2533 N1000 #46 Aug 28 2025 15:29:16
%S A002597 1,1,3,6,9,15,25,34,51,73,97,132,178,226,294,376,466,582,722,872,1062,
%T A002597 1282,1522,1812,2147,2507,2937,3422,3947,4557,5243,5978,6825,7763,
%U A002597 8771,9912,11172,12516,14028,15680,17444,19404,21540,23808,26316,29028,31908
%N A002597 Number of partitions into one kind of 1's, two kinds of 2's, and three kinds of 3's.
%C A002597 Old name was: A generalized partition function.
%D A002597 Gupta, Hansraj; A generalization of the partition function. Proc. Nat. Inst. Sci. India 17, (1951). 231-238.
%D A002597 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002597 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002597 Alois P. Heinz, <a href="/A002597/b002597.txt">Table of n, a(n) for n = 0..1000</a>
%H A002597 H. Gupta, <a href="/A002597/a002597.pdf">A generalization of the partition function</a>, Proc. Nat. Inst. Sci. India 17, (1951). 231-238. [Annotated scanned copy]
%H A002597 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1, 2, 1, -4, -5, 3, 6, 3, -5, -4, 1, 2, 1, -1).
%F A002597 G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)^3). - _Henry Bottomley_, Sep 17 2001
%F A002597 Euler transform of [1, 2, 3, 0, 0, 0, 0, 0, ...]. - _Thomas Wieder_, Mar 13 2005
%F A002597 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
%p A002597 a:= proc(n) option remember;
%p A002597      `if`(n=0, 1, add(add(d *`if`(d<4, d, 0),
%p A002597       d=numtheory[divisors](j)) *a(n-j), j=1..n)/n)
%p A002597     end:
%p A002597 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 21 2012
%t A002597 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_ *)
%t A002597 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 *)
%o A002597 (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
%Y A002597 Cf. A064349.
%K A002597 nonn,changed
%O A002597 0,3
%A A002597 _N. J. A. Sloane_
%E A002597 More terms from _Henry Bottomley_, Sep 17 2001
%E A002597 Better name from _Joerg Arndt_, Oct 12 2012