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 A055922 Number of partitions of n in which each part occurs an odd number (or zero) times. 14
 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-François Alcover, Feb 24 2015, after Alois P. Heinz *) PROG (PARI) { my(n=60); Vec(prod(k=1, n, 1 + sum(r=0, n\(2*k), x^(k*(2*r+1))) + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017 CROSSREFS Cf. A000041, A007690, A055923, A249389. Column k=0 of A264399. Sequence in context: A303764 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 April 7 13:36 EDT 2020. Contains 333305 sequences. (Running on oeis4.)