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A100824 Number of partitions of n with at most one odd part. 2


%S 1,1,1,2,2,4,3,7,5,12,7,19,11,30,15,45,22,67,30,97,42,139,56,195,77,

%T 272,101,373,135,508,176,684,231,915,297,1212,385,1597,490,2087,627,

%U 2714,792,3506,1002,4508,1255,5763,1575,7338,1958,9296,2436,11732,3010,14742

%N Number of partitions of n with at most one odd part.

%H Alois P. Heinz, <a href="/A100824/b100824.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1+x/(1-x^2))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).

%F a(n) ~ exp(sqrt(n/3)*Pi) / (2*sqrt(3)*n) if n is even and a(n) ~ exp(sqrt(n/3)*Pi) / (2*Pi*sqrt(n)) if n is odd. - _Vaclav Kotesovec_, Mar 07 2016

%p seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i),i=1..100),x,100),polynom),x,n),n=0..60); (C. Ronaldo)

%t nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 07 2016 *)

%Y Cf. A000041, A000070, A008951, A000097, A000098, A000710.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Jan 13 2005

%E More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

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Last modified July 16 11:09 EDT 2020. Contains 335784 sequences. (Running on oeis4.)