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

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, 108, 126, 146, 168, 194, 224, 256, 294, 336, 384, 439, 500, 568, 646, 732, 828, 938, 1060, 1194, 1348, 1516, 1704, 1916, 2149, 2408, 2698, 3018, 3372, 3766, 4202, 4682

Offset

0,4

Comments

Fine's numbers L(n).

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

References

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

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

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

(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]

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

a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 09 2019

Example

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

Maple

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
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, b(n, i-1)+
      `if`(i=n and i::odd, 1, 0)+`if`(i<n, b(n-i, min(n-i, i-1)), 0))
    end:
a:= n-> `if`(n=0, 0, b(n$2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 01 2019

Mathematica

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

Prog

(Haskell)
a026832 n = p 1 n where
   p _ 0 = 1
   p k m = if m < k then 0 else p (k+1) (m-k) + p (k+1+0^(n-m)) m
-- Reinhard Zumkeller, Jun 14 2012

Crossrefs

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

Cf. A000009, A096765.

Sequence in context: A163373 A117193 A325707 * A225044 A325246 A193691

Adjacent sequences: A026829 A026830 A026831 * A026833 A026834 A026835

Keyword

nonn,nice

Author

Clark Kimberling

Extensions

More terms from Emeric Deutsch, Mar 29 2006

a(0)=0 prepended by Alois P. Heinz, Feb 01 2019

Status

approved