A117276 - OEIS

%I #9 May 15 2018 20:43:17

%S 0,1,2,4,7,11,17,26,38,54,76,105,143,193,257,339,444,576,742,950,1208,

%T 1528,1923,2407,2999,3721,4597,5657,6937,8476,10322,12532,15168,18306,

%U 22034,26450,31672,37835,45091,53619,63625,75341,89037,105023,123647

%N Number of 1's in all partitions of n with no even parts repeated.

%C a(n)=Sum(k*A117274(n,k),k=0..n).

%F G.f.: x*product((1+x^(2j))/(1-x^(2j-1)), j=1..infinity)/(1-x).

%F a(n) ~ exp(sqrt(n/2)*Pi) / (2^(5/4)*Pi*n^(1/4)). - _Vaclav Kotesovec_, Mar 07 2016

%F G.f.: (x/(1 - x))*Product_{k>=1} (1 - x^(4*k))/(1 - x^k). - _Ilya Gutkovskiy_, May 15 2018

%e a(5)=11 because the partitions of 5 with no even parts repeated are [5],[4,1],[3,2],[3,1,1],[2,1,1,1] and [1,1,1,1,1] and they have a total number 0+1+0+2+3+5=11 parts equal to 1.

%p g:=x*product((1+x^(2*j))/(1-x^(2*j-1)),j=1..35)/(1-x): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..47);

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

%Y Cf. A001935, A117274.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Mar 06 2006