Search: id:A080995
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%I A080995
%S A080995 1,1,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,
%T A080995 1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
%U A080995 1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0
%N A080995 Characteristic function of generalized pentagonal numbers.
%C A080995 Repeatedly [1,[0,]^2k,1,[0,]^k], k>=0; characteristic function of generalized 
               pentagonal numbers: a(A001318(n))=1, a(A118300(n))=0. - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006
%D A080995 P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and 
               New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 81, Article 
               331.
%H A080995 T. D. Noe, <a href="b080995.txt">Table of n, a(n) for n=0..1001</a>
%H A080995 <a href="Sindx_Ch.html#char_fns">Index entries for characteristic functions</
               a>
%H A080995 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A080995 G.f.: Sum x^(n*(3n+1)/2), n=-inf..inf [the exponents are the pentagonal 
               numbers, A000326].
%F A080995 a(n)=b(24n+1) where b(n) is multiplicative and b(2^e)=b(3^e)=0^e, b(p^e)=(1+(-1)^e)/
               2 if p>3. - Michael Somos Jun 06 2005
%F A080995 Euler transform of period 6 sequence [ 1, 0, -1, 0, 1, -1, ...].
%F A080995 Expansion of phi(-q^3) / chi(-q) in powers of q where phi(), chi() are 
               Ramanujan theta functions. - Michael Somos Sep 14 2007
%F A080995 Expansion of psi(q) - q * psi(q^9) in powers of q^3 where psi() is a 
               Ramanujan theta function. - Michael Somos Sep 14 2007
%F A080995 Expansion of f(x, x^2) in powers of x where f() is Ramanujan's two-variable 
               theta function.
%F A080995 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 2^(1/
               2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for 
               A089810.
%F A080995 Expansion of q^(-1/24) * eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)) 
               in powers of q.
%F A080995 G.f.: Product_{k>0} (1 - x^(3*k)) / (1 - x^k + x^(2*k)). - Michael Somos 
               Jan 26 2008
%e A080995 q + q^25 + q^49 + q^121 + q^169 + q^289 + q^361 + q^529 + q^625 + ...
%o A080995 (PARI) a(n)=if(n<0,0,abs(polcoeff(eta(x+x*O(x^n)),n)))
%o A080995 (PARI) a(n)=issquare(24*n+1) /* Michael Somos Apr 13 2005 */
%o A080995 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)^2/
               eta(x+A)/eta(x^6+A), n))}
%Y A080995 |A010815(n)| = a(n). A089806(2n) = a(n). A033683(24n+1) = a(n).
%Y A080995 Sequence in context: A115513 A133080 A010815 this_sequence A121373 A133985 
               A143062
%Y A080995 Adjacent sequences: A080992 A080993 A080994 this_sequence A080996 A080997 
               A080998
%K A080995 nonn,easy
%O A080995 0,1
%A A080995 Michael Somos, Feb 27, 2003

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