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Search: id:A007325
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| A007325 |
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G.f.: Product_{k>0} (1-x^{5k-1})*(1-x^{5k-4})/((1-x^{5k-2})*(1-x^{5k-3})).
(Formerly M0415)
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+0
11
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| 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, 7, -1, -7, 14, -16, 11, -1, -11, 20, -22, 16, -2, -15, 29, -33, 23, -2, -23, 41, -45, 32, -4, -30, 57, -64, 45, -4, -43, 78, -86, 60, -7, -57, 107, -119, 83, -8, -79, 143
(list; graph; listen)
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OFFSET
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0,11
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COMMENT
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Hauptmodul series for GAMMA(5).
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
G. E. Andrews, Simplicity and surprise in Ramanujan's "Lost" Notebook, Amer. Math. Monthly, 104 (No. 10, Dec. 1997), 918-925.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 81.
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162; see Eq. (6.4).
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.
G. S. Joyce, Exact results for the activity and thermal compressibility of the hard-hexagon model, J. Phys. A 21 (1988), L983-L988.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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Euler transform of period 5 sequence [ -1,1,1,-1,0,...] (=-A080891).
G.f.: (Sum (-1)^n x^((5n+3)n/2))/(Sum (-1)^n x^((5n+1)n/2)). - Michael Somos, Dec 13 2002
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
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
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
Power series expansion of Rogers-Ramanujan's continued fraction 1/ (1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/... )))).
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EXAMPLE
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q - q^6 + q^11 - q^21 + q^26 - q^31 + q^36 - q^46 + 2*q^51 - 3*q^56 + ...
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MAPLE
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product( (1-x^(5*k-1))*(1-x^(5*k-4))/((1-x^(5*k-2))*(1-x^(5*k-3))), k=1..60);
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PROGRAM
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(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))
(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))
(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))
(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))
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CROSSREFS
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Cf. A055101, A055102, A055103, A003823.
Sequence in context: A050075 A002120 A021435 this_sequence A056619 A165192 A104771
Adjacent sequences: A007322 A007323 A007324 this_sequence A007326 A007327 A007328
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KEYWORD
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sign,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)
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