Search: id:A001936
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%I A001936 M1372 N0532
%S A001936 1,2,5,10,18,32,55,90,144,226,346,522,777,1138,1648,2362,3348,4704,6554,
%T A001936 9056,12425,16932,22922,30848,41282,54946,72768,95914,125842,164402,
%U A001936 213901,277204,357904,460448,590330,754368,960948,1220370,1545306
%N A001936 Expansion of q^(-1/4)(eta(q^4)/eta(q))^2 in powers of q.
%D A001936 A. Cayley, A memoir on the transformation of elliptic functions, Collected 
               Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, 
               Vol. 9, p. 128.
%D A001936 H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for 
               the integer coefficients of the theta function nome, Math. Comp., 
               29 (1975), 851-855.
%D A001936 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; Eq. (34.3).
%D A001936 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001936 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001936 T. D. Noe, <a href="b001936.txt">Table of n, a(n) for n=0..1000</a>
%F A001936 G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....
%F A001936 G.f.: eta(q^4)^2/(eta(q)^2*q^(1/4)) where eta = Dedekind's function.
%F A001936 A079006(n) = (-1)^n a(n).
%F A001936 Expansion of q^(-1/4)(1/2)(k/k')^(1/2) in powers of q.
%F A001936 Euler transform of period 4 sequence [2, 2, 2, 0, ...].
%F A001936 Given g.f. A(x), then B(x)=(x*A(x^4))^4 satisfies 0=f(B(x), B(x^2)) where 
               f(u, v)=(1+16u)(1+16v)v-u^2 . - Michael Somos Jul 09 2005
%F A001936 Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where 
               f(u, v)=(u^2+v^2)^2 -uv(1+4uv)^2 . - Michael Somos Jul 09 2005
%F A001936 G.f.: (Product_{k>0} (1+x^(2k))/(1-x^(2k-1)))^2 = (Product_{k>0} (1-x^(4k))/
               (1-x^k))^2 = square of g.f. for A001935.
%p A001936 with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; 
               local d,j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), 
               j=1..n)/n fi end end: a:= etr (n-> [2,2,2,0] [modp(n-1,4)+1]): seq 
               (a(n), n=0..38); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), 
               Sep 08 2008]
%o A001936 (PARI) a(n)=if(n<0,0,polcoeff((eta(x^4+x*O(x^n))/eta(x+x*O(x^n)))^2,n))
%o A001936 (PARI) a(n)=if(n<0,0,polcoeff(prod(k=1,n,1/if(k%4,1-x^k,1),1+x*O(x^n))^2,
               n))
%Y A001936 See A127391, A127392, A079006 for other versions of this sequence.
%Y A001936 A079006(n)=(-1)^n a(n). Convolution square of A001935.
%Y A001936 Sequence in context: A006327 A103577 A079006 this_sequence A127297 A018739 
               A011893
%Y A001936 Adjacent sequences: A001933 A001934 A001935 this_sequence A001937 A001938 
               A001939
%K A001936 nonn
%O A001936 0,2
%A A001936 N. J. A. Sloane (njas(AT)research.att.com).

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