cp's OEIS Frontend

A092316 Sum of largest parts of all partitions of n into odd distinct parts.

%I A092316 #28 Jan 19 2022 18:16:42
%S A092316 1,0,3,3,5,5,7,12,14,16,18,27,29,33,42,55,59,65,78,95,110,118,137,167,
%T A092316 188,200,236,274,303,330,376,435,485,522,591,677,741,803,903,1022,
%U A092316 1115,1210,1345,1505,1650,1784,1964,2201,2393,2578,2843,3143,3409,3685,4034
%N A092316 Sum of largest parts of all partitions of n into odd distinct parts.
%H A092316 Alois P. Heinz, <a href="/A092316/b092316.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Seiichi Manyama)
%H A092316 Arnold Knopfmacher and Neville Robbins, <a href="http://www.plouffe.fr/OEIS/citations/robbinspart.pdf">Identities for the total number of parts in partitions of integers</a>, Util. Math. 67 (2005), 9-18.
%F A092316 G.f.: Sum_{n>=1} (2*n-1)*x^(2*n-1)*Product_{k=1..n-1} (1+x^(2*k-1)).
%F A092316 a(n) = 2 * A067619(n) - A000700(n). - _Seiichi Manyama_, Jan 19 2022
%e A092316 a(13) = 29 because the partitions of 13 into distinct odd parts are [13],[9,3,1] and [7,5,1], with sum of largest terms 13+9+7 = 29.
%p A092316 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1 or i^2<n,
%p A092316        0, b(n, i-1)+ (t-> `if`(t>n, 0, b(n-t, i-1)))(2*i-1) ))
%p A092316     end:
%p A092316 a:= n-> add(`if`(j::odd, j*b(n-j, (j-1)/2), 0), j=1..n):
%p A092316 seq(a(n), n=1..55);  # _Alois P. Heinz_, Jan 19 2022
%t A092316 nmax = 50; Rest[CoefficientList[Series[Sum[(2*k - 1)*x^(2*k - 1) * Product[1 + x^(2*j - 1), {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jun 28 2016 *)
%Y A092316 Cf. A000700, A005895, A067619, A092319, A116857.
%K A092316 easy,nonn
%O A092316 1,3
%A A092316 _Vladeta Jovovic_, Feb 15 2004
%E A092316 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004