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A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.

%I A100847 #18 Jan 03 2016 06:31:01
%S A100847 1,2,3,7,10,17,28,42,62,93,137,193,276,383,532,734,997,1342,1807,2400,
%T A100847 3177,4190,5478,7130,9245,11923,15305,19591,24957,31673,40075,50518,
%U A100847 63460,79523,99296,123664,153616,190271,235072,289776,356302,437107,535112,653626
%N A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.
%H A100847 Alois P. Heinz, <a href="/A100847/b100847.txt">Table of n, a(n) for n = 0..1000</a>
%F A100847 G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^i).
%F A100847 a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((2*Pi^2/3 + 8*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jan 03 2016
%e A100847 a(3) = 7 because we have 6, 42, 411, 33, 222, 21111 and 111111.
%p A100847 g:=product((1+x^i-x^(2*i))/(1-x^i),i=1..50): gser:=series(g,x=0,40): seq(coeff(gser,x,n),n=0..35); # _Emeric Deutsch_, Aug 25 2007
%p A100847 # second Maple program:
%p A100847 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A100847       add(`if`(irem(i+j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
%p A100847        +b(n, i-1)))
%p A100847     end:
%p A100847 a:= n-> b(2*n$2):
%p A100847 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 31 2014
%t A100847 nmax = 50; CoefficientList[Series[Product[(1+x^k-x^(2*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 03 2016 *)
%Y A100847 Cf. A055922, A117958, A130126, A131942, A102247, A263401.
%K A100847 easy,nonn
%O A100847 0,2
%A A100847 _Vladeta Jovovic_, Aug 16 2007
%E A100847 More terms from _Emeric Deutsch_, Aug 25 2007