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A055922 Number of partitions of n in which each part occurs an odd number (or zero) times. 12
1, 1, 1, 3, 2, 5, 6, 9, 9, 16, 20, 25, 32, 40, 54, 69, 84, 101, 136, 156, 202, 244, 306, 357, 448, 527, 652, 773, 944, 1103, 1346, 1574, 1885, 2228, 2640, 3106, 3684, 4302, 5052, 5931, 6924, 8079, 9416, 10958, 12718, 14824, 17078, 19820, 22860, 26433 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

F. C. Auluck, K. S. Singwi and B. K. Agarwala, On a new type of partition, Proc. Nat. Inst. Sci. India 16 (1950) 147-156.

Steven Finch, Integer Partitions, September 22, 2004, page 5. [Cached copy, with permission of the author]

FORMULA

EULER transform of b where b has g.f. Sum {k>0} c(k)*x^k/(1-x^k) where c is inverse EULER transform of characteristic function of odd numbers.

G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^(2*i)). - Vladeta Jovovic, Feb 03 2004

Asymptotics (Auluck, Singwi, Agarwala, 1950): a(n) ~ B/(2*Pi*n) * exp(2*B*sqrt(n)), where B = sqrt(Pi^2/12 + 2*log(phi)^2), where phi is the golden ratio. - Vaclav Kotesovec, Oct 27 2014

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(`if`(irem(j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)

       +b(n, i-1)))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[j, 2] == 0, 0, b[n-i*j, i-1]], {j, 1, n/i}] + b[n, i-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-Fran├žois Alcover, Feb 24 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A007690, A055923, A249389.

Column k=0 of A264399.

Sequence in context: A277680 A079973 A267150 * A194072 A194105 A194012

Adjacent sequences:  A055919 A055920 A055921 * A055923 A055924 A055925

KEYWORD

nonn

AUTHOR

Christian G. Bower, Jun 23 2000

STATUS

approved

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Last modified August 20 11:34 EDT 2017. Contains 290835 sequences.