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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
8
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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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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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B(q) = q -q^6 +q^11 -q^21 +q^26 -q^31 +q^36 -q^46 +2*q^51 +...
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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 A104771 A056888
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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njas, Mira Bernstein (mira(AT)math.berkeley.edu)
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