A117276 Number of 1's in all partitions of n with no even parts repeated.
0, 1, 2, 4, 7, 11, 17, 26, 38, 54, 76, 105, 143, 193, 257, 339, 444, 576, 742, 950, 1208, 1528, 1923, 2407, 2999, 3721, 4597, 5657, 6937, 8476, 10322, 12532, 15168, 18306, 22034, 26450, 31672, 37835, 45091, 53619, 63625, 75341, 89037, 105023, 123647
Offset: 0
Keywords
Examples
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.
Programs
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Maple
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);
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Mathematica
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 *)
Formula
G.f.: x*product((1+x^(2j))/(1-x^(2j-1)), j=1..infinity)/(1-x).
a(n) ~ exp(sqrt(n/2)*Pi) / (2^(5/4)*Pi*n^(1/4)). - Vaclav Kotesovec, Mar 07 2016
G.f.: (x/(1 - x))*Product_{k>=1} (1 - x^(4*k))/(1 - x^k). - Ilya Gutkovskiy, May 15 2018
Comments