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A079006
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Expansion of q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.
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13
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1, -2, 5, -10, 18, -32, 55, -90, 144, -226, 346, -522, 777, -1138, 1648, -2362, 3348, -4704, 6554, -9056, 12425, -16932, 22922, -30848, 41282, -54946, 72768, -95914, 125842, -164402, 213901, -277204, 357904, -460448, 590330, -754368, 960948, -1220370
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| a(n) = (2/n)*Sum_{k=1..n} (-1)^k*A046897(k)*a(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 24 2002
Expansion of q^(-1/4)(1/2)k^(1/2) in powers of q.
Expansion of (1/q)(1/2)(1-sqrt(k'))/(1+sqrt(k')) in powers of q^4.
Euler transform of period 4 sequence [ -2, 4, -2, 0, ...].
G.f. A(x) satisfies A(x)^2 = A(x^2) / (1 + 4 * x * A(x^2)^2). - Michael Somos Mar 19 2004
Given g.f. A(x), then B(x) = x * A(x^4) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 * (1 + 4 * v^2) - v. - Michael Somos Jul 09 2005
Given g.f. A(x), then B(x) = x * A(x^4) satisfies 0 = f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = u1*u3 * (u6 + u2)^2 - u2*u6. - Michael Somos Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k-1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - (-x)^k))^2.
Expansion of continued fraction 1 / (1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 /(1 - x^10 + (x^3 + x^9)^2 / ...))). - Michael Somos Sep 01 2005
Given g.f. A(x), then B(x) = 2 * x * A(x^4) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4 . - Michael Somos Jan 01 2006
Expansion of psi(q^2) / phi(q) = psi(q)^2 / phi(q)^2 = psi(q^2)^2 / psi(q)^2 = psi(-q)^2 / phi(-q^2)^2 = chi(-q)^2 / chi(-q^2)^4 = 1 / (chi(q)^2 * chi(-q^2)^2) = 1 / (chi(q)^4 * chi(-q)^2) = f(-q^4)^2 / f(q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Convolution inverse of A029839. Convolution square of A083365. a(n) = (-1)^n * A001936(n).
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EXAMPLE
| q - 2*q^5 + 5*q^9 - 10*q^13 + 18*q^17 - 32*q^21 + 55*q^25 - 90*q^29 + ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ Product[(1 + x^(k + 1)) / (1 + x^k), {k, 1, n, 2}]^2, {x, 0, n}] (* Michael Somos Jul 08 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ (m / 16 / q)^(1/4), {q, 0, n}]] (* Michael Somos Jul 08 2011 *)
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PROG
| (PARI) {a(n) = local(N, A); if( n<0, 0, N = (sqrtint(16*n + 1) + 1)\2; A = contfracpnqn( matrix(2, N, i, j, if( i==1, if( j<2, 1 + O(x^(N^2 + N)), (x^(j-1) + x^(3*j - 3))^2), 1 - x^(4*j - 2)))); polcoeff( A[2, 1] / A[1, 1], 4*n))} /* Michael Somos Sep 01 2005 */
(PARI) {a(n) = local(A, m); if( n<0, 0, A = 1 + O(x); m = 1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A / (1 + 4 * x*A^2))); polcoeff(A, n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^2, n))}
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CROSSREFS
| Cf. A001936, A002103, A029839, A046897, A083365, A127391, A127392.
Sequence in context: A006327 A185721 A103577 * A001936 A127297 A018739
Adjacent sequences: A079003 A079004 A079005 * A079007 A079008 A079009
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KEYWORD
| sign,easy
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AUTHOR
| Michael Somos, Dec 22 2002
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