A026804 Number of partitions of n in which the least part is odd.
1, 1, 3, 3, 6, 8, 13, 16, 25, 33, 47, 61, 84, 109, 148, 189, 249, 319, 413, 522, 670, 842, 1066, 1330, 1668, 2068, 2574, 3171, 3915, 4800, 5888, 7175, 8753, 10617, 12879, 15552, 18772, 22570, 27125, 32480, 38867, 46372, 55275, 65707, 78047, 92470, 109456
Offset: 1
Keywords
Examples
a(5)=6 because we have [5],[4,1],[3,1,1],[2,2,1],[2,1,1,1] and [1,1,1,1,1] ([3,2] does not qualify).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Crossrefs
Cf. A046746.
Programs
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Maple
g:=sum(x^(2*k-1)/product(1-x^j,j=2*k-1..50),k=1..50): gser:=series(g,x=0,45): seq(coeff(gser,x,n),n=1..43); # Emeric Deutsch, Apr 04 2006 # second Maple program: b:= proc(n, i) option remember; `if`(n<1 or i<1, 0, b(n, i-1)+ `if`(n=i, irem(n, 2), 0)+`if`(i>n, 0, b(n-i, i))) end: a:= n-> b(n$2): seq(a(n), n=1..60); # Alois P. Heinz, Jul 26 2015
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Mathematica
b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, b[n, i - 1] + If[n == i, Mod[n, 2], 0] + If[i > n, 0, b[n - i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *) -
PARI
b(n, i) = if(n<1 || i<1, 0, b(n, i - 1) + if(n==i, n%2 , 0) + if(i>n, 0, b(n - i, i))); a(n) = b(n, n); \\ Indranil Ghosh, Jun 22 2017, after Maple code by Alois P. Heinz
Formula
G.f.: Sum_{k>=1}((-1)^(k+1)*(-1+1/Product_{i>=k} (1-x^i))). a(n) = Sum_{k=1..n}(-1)^(k+1)*A026807(n, k). - Vladeta Jovovic, Aug 26 2003
G.f.: Sum_{j>=1}(x^j/(1+x^j)/Product_{i=1..j}(1-x^i)). - Vladeta Jovovic, Aug 11 2004
G.f.: Sum_{k>=1}(x^(2k-1)/Product_{j>=2k-1}(1-x^j)). - Emeric Deutsch, Apr 04 2006
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6))) / sqrt(n) + (25/16 + 2929*Pi^2/6912)/n). - Vaclav Kotesovec, Jul 06 2019, extended Nov 02 2019
Comments