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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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12
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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
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OFFSET
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0,11
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COMMENTS
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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.
Hauptmodul series for GAMMA(5).
Expansion of Rogers-Ramanujan's continued fraction 1 / (1 + x / ( 1 + x^2 / ( 1 + x^3 / ( 1 + x^4 / ... )))).
Given the g.f. A(x) the notation R(q) := q^(1/5) * A(q) is used by Berndt.
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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.
G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005, see p. 57.
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 9.
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
P. J. Nahin, Number-Crunching, Princeton University Press, 2011. See p. 22 equation (2.2.4).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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
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Euler transform of period 5 sequence [ -1 , 1, 1, -1, 0, ...] (=-A080891).
G.f.: prod(k>=1, (1-x^(5*k-1)) * (1-x^(5*k-4)) / ( (1-x^(5*k-2)) * (1-x^(5*k-3)) ) ) = H(x) / G(x) where H and G are respectively the g.f. of A003114 and A003106.
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
G.f.: 1 / (1 + x / ( 1 + x^2 / ( 1 + x^3 / ( 1 + x^4 / ... )))).
G.f.: 1 / (1 + 1 / (x^-1 + 1 / (x^-1 + 1 / (x^-2 + 1 / (x^-2 + 1 / ... ))))). - Michael Somos, Apr 30 2012
G.f.: A(x) = S(0) -1; S(k) = 1 + x^k/S(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 18 2011
Hankel transform is A167683. - Michael Somos, Apr 30 2012
a(n) = (-1)^n * A226556(n). - Michael Somos, Jun 11 2013
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EXAMPLE
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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
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t1:=mul((1-x^(5*k-1))*(1-x^(5*k-4))/((1-x^(5*k-2))*(1-x^(5*k-3))), k=1..60); seriestolist(series(t1, x, 59)); - N. J. A. Sloane, Jun 10 2013
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
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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 *)
a[ n_] := SeriesCoefficient[ ContinuedFractionK[ q^k, 1, {k, 0, n}], {q, 0, n}] (* Michael Somos, Jun 10 2013 *)
max = 65; CoefficientList[ Series[ Fold[ #2/(1 + #1)&, q^n, q^Reverse[ Range[0, max-1] ] ], {q, 0, max}], q] (* Jean-François Alcover, Apr 04 2013 *)
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PROG
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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, A167683.
Sequence in context: A002120 A021435 A226556 * A187038 A056619 A165192
Adjacent sequences: A007322 A007323 A007324 * A007326 A007327 A007328
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KEYWORD
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sign,easy,nice,changed
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AUTHOR
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N. J. A. Sloane, Mira Bernstein (mira(AT)math.berkeley.edu)
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STATUS
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approved
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