Search: id:A007325
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%I A007325 M0415
%S A007325 1,1,1,0,1,1,1,1,0,1,2,3,2,0,2,4,4,3,1,3,6,7,5,0,5,9,10,
%T A007325 7,1,7,14,16,11,1,11,20,22,16,2,15,29,33,23,2,23,41,45,
%U A007325 32,4,30,57,64,45,4,43,78,86,60,7,57,107,119,83,8,79,143
%V A007325 1,-1,1,0,-1,1,-1,1,0,-1,2,-3,2,0,-2,4,-4,3,-1,-3,6,-7,5,0,-5,9,-10,
%W A007325 7,-1,-7,14,-16,11,-1,-11,20,-22,16,-2,-15,29,-33,23,-2,-23,41,-45,
%X A007325 32,-4,-30,57,-64,45,-4,-43,78,-86,60,-7,-57,107,-119,83,-8,-79,143
%N A007325 G.f.: Product_{k>0} (1-x^{5k-1})*(1-x^{5k-4})/((1-x^{5k-2})*(1-x^{5k-3})).
%C A007325 Hauptmodul series for GAMMA(5).
%D A007325 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007325 G. E. Andrews, Simplicity and surprise in Ramanujan's "Lost" Notebook, 
               Amer. Math. Monthly, 104 (No. 10, Dec. 1997), 918-925.
%D A007325 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 81.
%D A007325 W. Duke, Continued fractions and modular functions, Bull. Amer. Math. 
               Soc., 42 (2005), 137-162; see Eq. (6.4).
%D A007325 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 
               3, p. 24.
%D A007325 G. S. Joyce, Exact results for the activity and thermal compressibility 
               of the hard-hexagon model, J. Phys. A 21 (1988), L983-L988.
%H A007325 T. D. Noe, <a href="b007325.txt">Table of n, a(n) for n=0..1000</a>
%H A007325 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Rogers-RamanujanContinuedFraction.html">Link to a section of The 
               World of Mathematics.</a>
%F A007325 Euler transform of period 5 sequence [ -1,1,1,-1,0,...] (=-A080891).
%F A007325 G.f.: (Sum (-1)^n x^((5n+3)n/2))/(Sum (-1)^n x^((5n+1)n/2)). - Michael 
               Somos, Dec 13 2002
%F A007325 Given g.f. A(x), then B(x)=x*A(x^5) satisfies 0=f(B(x), B(x^2)) where 
               f(u, v)=u^2-v+u*v^3+u^3*v^2 . - Michael Somos Mar 09 2004
%F A007325 Given g.f. A(x), then B(x)=x*A(x^5) satisfies 0=f(B(x), B(x^2), B(x^4)) 
               where f(u, v, w)=u(uv+w^2+v^2w)-w . - Michael Somos Aug 29 2005
%F A007325 Given g.f. A(x), then B(x)=x*A(x^5) satisfies 0=f(B(x), B(x^2), B(x^3), 
               B(x^6)) where f(u1, u2, u3, u6)=u1*u2+u1*u3^2*u6+u2*u3^2-u2^2*u3*u6-u3 
               . - Michael Somos Aug 29 2005
%F A007325 Power series expansion of Rogers-Ramanujan's continued fraction 1/ (1+q/ 
               ( 1+q^2/ ( 1+q^3/ ( 1+q^4/... )))).
%e A007325 q - q^6 + q^11 - q^21 + q^26 - q^31 + q^36 - q^46 + 2*q^51 - 3*q^56 + 
               ...
%p A007325 product( (1-x^(5*k-1))*(1-x^(5*k-4))/((1-x^(5*k-2))*(1-x^(5*k-3))), k=1..60);
%o A007325 (PARI) a(n)=local(k); if(n<0,0,k=(3+sqrtint(9+40*n))\10; polcoeff(sum(n=-k,
               k,(-1)^n*x^((5*n^2+3*n)/2),x*O(x^n))/sum(n=-k,k,(-1)^n*x^((5*n^2+n)/
               2),x*O(x^n)),n))
%o A007325 (PARI) a(n)=if(n<0,0, polcoeff(prod(k=1,n,if(k%5,(1-x^k)^((-1)^binomial(k%5,
               2)),1),1+x*O(x^n)),n))
%o A007325 (PARI) a(n)=local(cf); if(n<0,0,cf=contfracpnqn(matrix(2,(sqrtint(8*n+1)+1)\2,
               i,j,if(i==1,x^(j-1),1))); polcoeff(cf[2,1]/cf[1,1]+x*O(x^n),n))
%o A007325 (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=5; A=x*subst(A,
               x,x^5); A=(A*(1-2*A+4*A^2-3*A^3+A^4)/(1+3*A+4*A^2+2*A^3+A^4)/x)^(1/
               5)); polcoeff(A,n))
%Y A007325 Cf. A055101, A055102, A055103, A003823.
%Y A007325 Sequence in context: A050075 A002120 A021435 this_sequence A056619 A165192 
               A104771
%Y A007325 Adjacent sequences: A007322 A007323 A007324 this_sequence A007326 A007327 
               A007328
%K A007325 sign,easy,nice
%O A007325 0,11
%A A007325 N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)

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