Search: id:A022567
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%I A022567
%S A022567 1,2,3,6,9,14,22,32,46,66,93,128,176,238,319,426,562,736,960,1242,1598,
%T A022567 2048,2608,3306,4175,5248,6570,8198,10190,12622,15589,19190,23552,28830,
%U A022567 35190,42842,52034,63040,76198,91904,110604,132832,159216,190464,227417
%N A022567 Expansion of Product (1+q^m)^2; m=1..inf.
%C A022567 Number of partitions of n into distinct parts, with 2 types of each part. 
               E.g. for n=4, we consider k and k* to be different versions of k 
               and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*, thus 
               a(4)=9 - Jon Perry (perry(AT)globalnet.co.uk), Apr 04 2004
%C A022567 Number of partitions of n into odd parts, each part being of two kinds. 
               E.g. a(3)=6 because we have 3, 3', 1+1+1, 1+1+1', 1+1'+1', 1'+1'+1'. 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2005
%C A022567 Euler transform of period 2 sequence [2,0,2,0,...]. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Mar 22 2005
%C A022567 Equals A000041 convolved with A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 11 2009]
%H A022567 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=852">
               Encyclopedia of Combinatorial Structures 852</a>
%F A022567 a(n) = p(n)+p(n-1)+p(n-3)+p(n-6)+...+p(n-k*(k+1)/2)+..., where p() is 
               A000041(). E.g. a(8) = p(8)+p(7)+p(5)+p(2) = 22+15+7+2 = 46. - Vladeta 
               Jovovic (vladeta(AT)eunet.rs), Aug 09 2004
%F A022567 Expansion of q^(-1/12) * (eta(q^2) / eta(q))^2 in powers of q.
%F A022567 Expansion of chi(-q)^(-2) in powers of q where chi() is a Ramanujan theta 
               function.
%F A022567 G.f. is a period 1 Fourier series which satisfies f(-1/(288 t)) = (1/
               2) / f(t) where q = exp(2 pi i t).
%F A022567 Parity result: a(n) is even except when n is twice a generalised pentagonal 
               number (i.e. of the form 2*A001318(m) for some m). [From Peter Bala 
               (pbala(AT)talktalk.net), Mar 19 2009]
%e A022567 q + 2*q^13 + 3*q^25 + 6*q^37 + 9*q^49 + 14*q^61 + 22*q^73 + 32*q^85 + 
               ...
%o A022567 (PARI) a(n)=if(n<0,0,polcoeff(prod(k=1,n,1+x^k,1+x*O(x^n))^2,n))
%o A022567 (PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A))^2, 
               n))
%Y A022567 Convolution square of A000009. Convolution inverse of A022597.
%Y A022567 A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]
%Y A022567 Sequence in context: A094056 A058609 A128518 this_sequence A134004 A123631 
               A018060
%Y A022567 Adjacent sequences: A022564 A022565 A022566 this_sequence A022568 A022569 
               A022570
%K A022567 nonn
%O A022567 0,2
%A A022567 N. J. A. Sloane (njas(AT)research.att.com).

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