cp's OEIS Frontend

A102247 Number of partitions of n in which each odd part has odd multiplicity and each even part has even multiplicity.

%I A102247 #14 Jan 03 2016 09:06:10
%S A102247 1,1,0,2,2,3,2,4,7,8,8,10,17,17,20,26,39,39,46,56,77,85,96,116,154,
%T A102247 172,190,234,289,328,364,440,532,610,670,808,957,1091,1204,1432,1675,
%U A102247 1905,2110,2476,2867,3255,3608,4184,4837,5451,6050,6960,7980,8961,9972,11370
%N A102247 Number of partitions of n in which each odd part has odd multiplicity and each even part has even multiplicity.
%H A102247 Alois P. Heinz, <a href="/A102247/b102247.txt">Table of n, a(n) for n = 0..1000</a>
%F A102247 G.f.: Product_{i>0} (1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)).
%F A102247 a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((Pi^2/3 + 4*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jan 03 2016
%e A102247 a(7) = 4 because we have 7, 322, 22111 and 1111111.
%p A102247 g:=product((1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)),i=1..40): gser:=series(g,x=0, 60): seq(coeff(gser,x,n),n=0..55); # _Emeric Deutsch_, Aug 23 2007
%p A102247 # second Maple program:
%p A102247 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A102247       add(`if`(irem(i+j, 2)=0, b(n-i*j, i-1), 0), j=1..n/i)
%p A102247        +b(n, i-1)))
%p A102247     end:
%p A102247 a:= n-> b(n$2):
%p A102247 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 31 2014
%t A102247 nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 03 2016 *)
%Y A102247 Cf. A055922, A117958, A130126, A131942, A100847.
%K A102247 easy,nonn
%O A102247 0,4
%A A102247 _Vladeta Jovovic_, Aug 16 2007
%E A102247 More terms from _Emeric Deutsch_, Aug 23 2007