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A026804 Number of partitions of n in which the least part is odd. 29
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Also number of partitions of n in which the largest part occurs an odd number of times. Example: a(5)=6 because we have [5],[4,1],[3,2],[3,1,1],[2,1,1,1] and [1,1,1,1,1] ([2,2,1] does not qualify). - Emeric Deutsch, Apr 04 2006
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from 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
EXAMPLE
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).
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
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 *)
PROG
(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
CROSSREFS
Cf. A046746.
Sequence in context: A333526 A097307 A323435 * A240213 A205970 A104715
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified April 13 21:47 EDT 2024. Contains 371645 sequences. (Running on oeis4.)