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A007325 G.f.: Product_{k>0} (1-x^{5k-1})*(1-x^{5k-4})/((1-x^{5k-2})*(1-x^{5k-3})).
(Formerly M0415)
12
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; text; internal format)
OFFSET

0,11

COMMENTS

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.

REFERENCES

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.

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).

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

J. Malenfant, Generalizing Ramanujan's J Functions, arXiv preprint arXiv:1109.5957, 2011

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.

FORMULA

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

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 + ...

MAPLE

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

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 *)

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-Fran├žois Alcover, Apr 04 2013 *)

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))}

CROSSREFS

Cf. A055101, A055102, A055103, A003823, A167683.

Sequence in context: A002120 A021435 A226556 * A247920 A187038 A056619

Adjacent sequences:  A007322 A007323 A007324 * A007326 A007327 A007328

KEYWORD

sign,easy,nice,changed

AUTHOR

N. J. A. Sloane, Mira Bernstein

STATUS

approved

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Last modified October 31 10:58 EDT 2014. Contains 248861 sequences.