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Search: id:A078905
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| A078905 |
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The q expansion of Gamma(5)-modular function Lambda^5. |
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+0
5
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| 1, -5, 15, -30, 40, -26, -30, 125, -220, 245, -124, -180, 615, -1010, 1085, -550, -705, 2415, -3850, 3980, -1926, -2460, 8090, -12550, 12715, -6074, -7500, 24360, -37150, 36930, -17251, -21155, 67380, -101210, 99295, -45924, -55305, 174500, -259140, 251275, -114750
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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G.f. A(x) satisfies 0=f(A(x),A(x^2)) where f(u,v)=u^2-v+u*v^3+u^3*v^2+10*u*v*(1-u+v+u*v). - Michael Somos Mar 09 2004
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REFERENCES
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A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 12, Entry 1(ii).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1001
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FORMULA
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G.f.: x(Prod_{k>0} (1-x^{5k-1})(1-x^{5k-4})/(1-x^{5k-2})(1-x^{5k-3}))^5
G.f.: x((Sum (-1)^n x^((5n+3)n/2))/(Sum (-1)^n x^((5n+1)n/2)))^5.
G.f. A(x)=x*B(x)^5 where B(x) is g.f. of A007325.
Euler transform of period 5 sequence [ -5, 5, 5, -5, 0, ...].
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EXAMPLE
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Lambda^5 = q -5q^2 +15q^3 -30q^4 +40q^5 +... where q=e^(2Pi i t).
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PROGRAM
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(PARI) a(n)=local(k); if(n<1, 0, k=(7+sqrtint(40*n-32))\10; polcoeff(x*(sum(i=-k, k, (-1)^i*x^((5*i^2+3*i)/2), O(x^n))/sum(i=-k, k, (-1)^i*x^((5*i^2+i)/2), O(x^n)))^5, n))
(PARI) a(n)=local(A); if(n<1, 0, A=O(x^n); A=(eta(x+A)/eta(x^5+A))^6/x; polcoeff(2/(11+A+sqrt(125+22*A+A^2)), n))
(PARI) {a(n)=local(A, u, v); if(n<0, 0, A=x; for(k=2, n, u=A+x*O(x^k); v=subst(u, x, x^2); A-=x^k*polcoeff(u^2-v+u*v^3+u^3*v^2+10*u*v*(1-u+v+u*v), k+1)/2); polcoeff(A, n))}
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CROSSREFS
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Cf. A007325.
Sequence in context: A076843 A129393 A162525 this_sequence A059160 A028895 A010898
Adjacent sequences: A078902 A078903 A078904 this_sequence A078906 A078907 A078908
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KEYWORD
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sign,easy,nice
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AUTHOR
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Michael Somos, Dec 12 2002
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