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A035457 Number of partitions of n into parts of the form 4*k + 2.

%I A035457 #64 Jul 28 2025 20:17:29
%S A035457 1,0,1,0,1,0,2,0,2,0,3,0,4,0,5,0,6,0,8,0,10,0,12,0,15,0,18,0,22,0,27,
%T A035457 0,32,0,38,0,46,0,54,0,64,0,76,0,89,0,104,0,122,0,142,0,165,0,192,0,
%U A035457 222,0,256,0,296,0,340,0,390,0,448,0,512,0,585,0,668,0,760,0,864,0,982,0
%N A035457 Number of partitions of n into parts of the form 4*k + 2.
%C A035457 Also number of partitions of n into distinct even parts. Example: a(10)=3 because we have [10],[8,2] and [6,4]. - _Emeric Deutsch_, Feb 22 2006
%C A035457 Also number of partitions of n into odd parts, each part occurring an even number of times. Example: a(10)=3 because we have [5,5], [3,3,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1]. - _Emeric Deutsch_, Apr 08 2006
%D A035457 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 288-289.
%H A035457 Vincenzo Librandi, <a href="/A035457/b035457.txt">Table of n, a(n) for n = 0..1000</a>
%F A035457 G.f.: 1/Product_{n>=0} (1 - x^(4*n+2)).
%F A035457 G.f.: 1/Product_{j>=0} (1 - x^(8*j+2))*(1 - x^(8*j+6)).
%F A035457 G.f.: Product_{j>=1} (1 + x^(2*j)). - _Emeric Deutsch_, Feb 22 2006
%F A035457 a(2*n-1) = 0, a(2*n) = A000009(n). a(n) = A116675(n,0). - _Emeric Deutsch_, Feb 22 2006
%F A035457 G.f.: Sum_{n>=1} (x^(n*(n+1)) / Product_{k=1..n} (1 - x^(2*k))). - _Joerg Arndt_, Mar 10 2011
%F A035457 If n is even, a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Feb 26 2015
%F A035457 a(4*n) = A035294(n) and a(4*n+2) = A078408(n). - _George Beck_, Aug 19 2017
%e A035457 a(10)=3 because we have [10], [6,2,2] and [2,2,2,2,2].
%p A035457 g:=product(1+x^(2*j),j=1..45): gser:=series(g,x=0,85): seq(coeff(gser,x,n),n=0..79); # _Emeric Deutsch_, Feb 22 2006; a(0) added by _Georg Fischer_, Dec 10 2020
%t A035457 nn=80;CoefficientList[Series[Product[1+ x^(2i),{i,1,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Jun 20 2014 *)
%t A035457 nmax = 50; kmax = nmax/4; s = Range[0, kmax]*4 + 2;
%t A035457 Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* _Robert Price_, Aug 03 2020 *)
%o A035457 (PARI)
%o A035457 N=166;  S=2+sqrtint(N);  x='x+O('x^N);
%o A035457 gf=sum(n=0, S, x^(n^2+n)/prod(k=1,n, 1-x^(2*k)) );
%o A035457 Vec(gf)
%o A035457 \\ _Joerg Arndt_, Feb 18 2014
%Y A035457 Cf. A000726, A116675.
%K A035457 nonn
%O A035457 0,7
%A A035457 _Olivier Gérard_
%E A035457 More terms from _Emeric Deutsch_, Feb 22 2006
%E A035457 Description simplified by _Joerg Arndt_, Jun 24 2009
%E A035457 a(0)=1 prepended by _Joerg Arndt_, Mar 11 2011