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A007325
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Expansion of f(-x, -x^4) / f(-x^2, -x^3) in powers of x where f(,) is Ramanujan's two variable theta function.
(Formerly M0415)
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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; refs; listen; history; internal format)
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OFFSET
| 0,11
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COMMENTS
| Hauptmodul series for GAMMA(5).
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REFERENCES
| 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.
J. Malenfant, Generalizing Ramanujan's J Functions, Arxiv preprint arXiv:1109.5957, 2011
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Rogets-Ramanujan Continued Fraction.
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FORMULA
| Euler transform of period 5 sequence [ -1 , 1, 1, -1, 0, ...] (=-A080891).
G.f.: (Sum (-1)^k x^((5*k + 3)*k/2))/(Sum (-1)^k x^((5*k + 1)*k/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 * (u*v + w^2 + v^2*w) - 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 + x / ( 1 + x^2 / ( 1 + x^3 / ( 1 + x^4 /... )))) in powers of x.
G.f.: Product_{k>0} (1 - x^{5*k - 1}) * (1 - x^{5*k - 4}) / ((1 - x^{5*k - 2}) * (1 - x^{5*k - 3})).
G.f.: A(x) = S(0) -1; S(k) = 1 + x^k/S(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 18 2011
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EXAMPLE
| 1 - x + x^2 - x^4 + x^5 - x^6 + x^7 - x^9 + 2*x^10 - 3*x^11 + 2*x^12 - ...
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
| product( (1-x^(5*k-1))*(1-x^(5*k-4))/((1-x^(5*k-2))*(1-x^(5*k-3))), k=1..60);
A007325_G:=proc(x, NK); Digits:=250;
Q2:=1;
for k from NK by -1 to 0 do
Q1:=1+x^k/Q2; Q2:=Q1; od;
Q3:=Q2; S:=Q3-1;
end;
- Sergei N. Gladkovskii, Dec 18 2011
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^5] QPochhammer[ q^4, q^5] / (QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5]), {q, 0, n}] (* Michael Somos, Aug 17 2011 *)
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PROG
| (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
| Cf. A055101, A055102, A055103, A003823.
Sequence in context: A050075 A002120 A021435 * A187038 A056619 A165192
Adjacent sequences: A007322 A007323 A007324 * A007326 A007327 A007328
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KEYWORD
| sign,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)
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