A026832 - OEIS

%I #36 May 14 2022 13:38:53

%S 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 108,126,146,168,194,224,256,294,336,384,439,500,568,646,732,828,938,

%U 1060,1194,1348,1516,1704,1916,2149,2408,2698,3018,3372,3766,4202,4682

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

%C Fine's numbers L(n).

%C 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 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).

%H Alois P. Heinz, <a href="/A026832/b026832.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: Sum_{k>=1} ((-1)^(k+1)*(-1+Product_{i>=k} (1+x^i))). - _Vladeta Jovovic_, Aug 26 2003

%F 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 (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 G.f.: Sum_{k>=1} x^(2k-1)*Product_{j>=2k} (1 + x^j). - _Emeric Deutsch_, Mar 29 2006

%F a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 09 2019

%e a(7)=4 because we have [7], [6,1], [4,3] and [4,2,1].

%p 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 # second Maple program:

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, b(n, i-1)+

%p       `if`(i=n and i::odd, 1, 0)+`if`(i<n, b(n-i, min(n-i, i-1)), 0))

%p     end:

%p a:= n-> `if`(n=0, 0, b(n$2)):

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 01 2019

%t 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 Join[{0},Table[Length[Select[IntegerPartitions[n],OddQ[#[[-1]]]&&Max[Tally[#][[All,2]]] == 1&]],{n,60}]] (* _Harvey P. Dale_, May 14 2022 *)

%o (Haskell)

%o a026832 n = p 1 n where

%o    p _ 0 = 1

%o    p k m = if m < k then 0 else p (k+1) (m-k) + p (k+1+0^(n-m)) m

%o -- _Reinhard Zumkeller_, Jun 14 2012

%Y Cf. A026804, A026805, A026807, A026833, A092265, A096661, A097042.

%Y Cf. A000009, A096765.

%K nonn,nice

%O 0,4

%A _Clark Kimberling_

%E More terms from _Emeric Deutsch_, Mar 29 2006

%E a(0)=0 prepended by _Alois P. Heinz_, Feb 01 2019