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A002134 Generalized divisor function. Number of partitions of n with exactly three part sizes.

%I A002134 M1367 N0530 #41 Sep 15 2023 18:45:54
%S A002134 1,2,5,10,15,25,37,52,67,97,117,154,184,235,277,338,385,469,531,630,
%T A002134 698,810,910,1038,1144,1295,1425,1577,1741,1938,2089,2301,2505,2700,
%U A002134 2970,3189,3444,3703,4004,4242,4617,4882,5244,5558,5999,6221,6755,7050,7576
%N A002134 Generalized divisor function. Number of partitions of n with exactly three part sizes.
%D A002134 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002134 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002134 Alois P. Heinz, <a href="/A002134/b002134.txt">Table of n, a(n) for n = 6..10000</a>
%H A002134 P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
%F A002134 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
%e A002134 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.
%p A002134 # Using function P from A365676:
%p A002134 A002134 := n -> P(n, 3, n): seq(A002134(n), n = 6..54); # _Peter Luschny_, Sep 15 2023
%t A002134 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 *)
%Y A002134 A diagonal of A060177.
%Y A002134 Column k=3 of A116608. - _Alois P. Heinz_, Nov 07 2012
%K A002134 nonn,easy
%O A002134 6,2
%A A002134 _N. J. A. Sloane_
%E A002134 Better description and more terms from _Naohiro Nomoto_, Jan 24 2002
%E A002134 More terms from _Vladeta Jovovic_, Nov 02 2003