cp's OEIS Frontend

A026832 Number of partitions of n into distinct parts, the least being odd.

%I A026832 #36 May 14 2022 13:38:53
%S A026832 0,1,0,2,1,2,2,4,4,5,6,8,10,12,14,18,21,24,30,36,42,50,58,68,80,93,
%T A026832 108,126,146,168,194,224,256,294,336,384,439,500,568,646,732,828,938,
%U A026832 1060,1194,1348,1516,1704,1916,2149,2408,2698,3018,3372,3766,4202,4682
%N A026832 Number of partitions of n into distinct parts, the least being odd.
%C A026832 Fine's numbers L(n).
%C A026832 Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(7)=4 because we have [3,2,1,1], [2,2,2,1], [2,1,1,1,1,1] and [1,1,1,1,1,1,1]. - _Emeric Deutsch_, Mar 29 2006
%D A026832 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).
%H A026832 Alois P. Heinz, <a href="/A026832/b026832.txt">Table of n, a(n) for n = 0..10000</a>
%F A026832 G.f.: Sum_{k>=1} ((-1)^(k+1)*(-1+Product_{i>=k} (1+x^i))). - _Vladeta Jovovic_, Aug 26 2003
%F A026832 G.f.: Sum_{ k >= 1 } x^(k*(k+1)/2)/((1+x^k)*Product_{i=1..k} (1-x^i) ). - _Vladeta Jovovic_, Aug 10 2004
%F A026832 (1 + Sum_{n >= 1} a(n)q^n )*(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)) = Sum_{n >= 1} (-1)^n*q^((3*n^2+n)/2)/(1+q^n). [Fine]
%F A026832 G.f.: Sum_{k>=1} x^(2k-1)*Product_{j>=2k} (1 + x^j). - _Emeric Deutsch_, Mar 29 2006
%F A026832 a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 09 2019
%e A026832 a(7)=4 because we have [7], [6,1], [4,3] and [4,2,1].
%p A026832 g:=sum(x^(2*k-1)*product(1+x^j, j=2*k..60), k=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..53); # _Emeric Deutsch_, Mar 29 2006
%p A026832 # second Maple program:
%p A026832 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, b(n, i-1)+
%p A026832       `if`(i=n and i::odd, 1, 0)+`if`(i<n, b(n-i, min(n-i, i-1)), 0))
%p A026832     end:
%p A026832 a:= n-> `if`(n=0, 0, b(n$2)):
%p A026832 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 01 2019
%t A026832 mx=53; Rest[CoefficientList[Series[Sum[x^(2*k-1) Product[1+x^j, {j, 2*k, mx}], {k, mx}], {x, 0, mx}], x]]  (* _Jean-François Alcover_, Apr 05 2011, after _Emeric Deutsch_ *)
%t A026832 Join[{0},Table[Length[Select[IntegerPartitions[n],OddQ[#[[-1]]]&&Max[Tally[#][[All,2]]] == 1&]],{n,60}]] (* _Harvey P. Dale_, May 14 2022 *)
%o A026832 (Haskell)
%o A026832 a026832 n = p 1 n where
%o A026832    p _ 0 = 1
%o A026832    p k m = if m < k then 0 else p (k+1) (m-k) + p (k+1+0^(n-m)) m
%o A026832 -- _Reinhard Zumkeller_, Jun 14 2012
%Y A026832 Cf. A026804, A026805, A026807, A026833, A092265, A096661, A097042.
%Y A026832 Cf. A000009, A096765.
%K A026832 nonn,nice
%O A026832 0,4
%A A026832 _Clark Kimberling_
%E A026832 More terms from _Emeric Deutsch_, Mar 29 2006
%E A026832 a(0)=0 prepended by _Alois P. Heinz_, Feb 01 2019