cp's OEIS Frontend

A026833 Number of partitions of n into distinct parts, the least being even.

%I A026833 #27 May 16 2025 02:20:13
%S A026833 0,0,1,0,1,1,2,1,2,3,4,4,5,6,8,9,11,14,16,18,22,26,31,36,42,49,57,66,
%T A026833 76,88,102,116,134,154,176,201,229,260,296,336,381,432,488,550,622,
%U A026833 700,788,886,994,1115,1250,1399,1564,1748,1952,2176,2426,2701,3004
%N A026833 Number of partitions of n into distinct parts, the least being even.
%C A026833 Also number of partitions of n such that if k is the largest part, then k occurs an even number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(10)=4 because we have [3,3,2,1,1], [2,2,2,2,1,1], [2,2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1]. - _Emeric Deutsch_, Mar 30 2006
%H A026833 Alois P. Heinz, <a href="/A026833/b026833.txt">Table of n, a(n) for n = 0..10000</a>
%F A026833 G.f.: Sum_{k>=2} ((-1)^k*(-1+Product_{i>=k} (1+x^i))). - _Vladeta Jovovic_, Aug 26 2003
%F A026833 G.f.: Sum_{k>=1} x^(2k)*Product_{j>=2k+1} (1+x^j).
%F A026833 G.f.: Sum_{k>=1} x^(k*(k+3)/2)/((1+x^k)*Product_{j=1..k} (1-x^j)). - _Emeric Deutsch_, Mar 30 2006
%F A026833 a(n) ~ exp(Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 09 2019
%e A026833 a(10)=4 because we have [10], [8,2], [6,4] and [5,3,2].
%p A026833 g:=sum(x^(2*k)*product(1+x^j, j=2*k+1..60), k=1..60): gser:=series(g, x=0, 58): seq(coeff(gser, x, n), n=0..55); # _Emeric Deutsch_, Mar 30 2006
%p A026833 # second Maple program:
%p A026833 b:= proc(n, i) option remember; `if`(i*(i+1)/2-1<n, 0, b(n, i-1)+
%p A026833       `if`(i=n and i::even, 1, 0)+`if`(i<n, b(n-i, min(n-i, i-1)), 0))
%p A026833     end:
%p A026833 a:= n-> b(n$2):
%p A026833 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 01 2019
%t A026833 b[n_, i_] := b[n, i] = If[i*(i+1)/2-1 < n, 0, b[n, i-1] + If[i == n && EvenQ[i], 1, 0]+If[i < n, b[n-i, Min[n-i, i-1]], 0]];
%t A026833 a[n_] := b[n, n];
%t A026833 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 16 2025, after _Alois P. Heinz_ *)
%Y A026833 Cf. A026832.
%K A026833 nonn
%O A026833 0,7
%A A026833 _Clark Kimberling_
%E A026833 a(0)=0 prepended by _Alois P. Heinz_, Feb 01 2019