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Search: id:A081360
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| A081360 |
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Expansion of q^(-1/24)(m(1-m)/16)^(1/24) in powers of q, where m=k^2 is the parameter and q is the nome for Jacobian elliptic functions. |
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3
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| 1, -1, 1, -2, 2, -3, 4, -5, 6, -8, 10, -12, 15, -18, 22, -27, 32, -38, 46, -54, 64, -76, 89, -104, 122, -142, 165, -192, 222, -256, 296, -340, 390, -448, 512, -585, 668, -760, 864, -982, 1113, -1260, 1426, -1610, 1816, -2048, 2304, -2590, 2910, -3264, 3658, -4097, 4582, -5120, 5718, -6378
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Number of partitions of n into distinct parts with an even number of odd parts minus partitions of n into distinct parts with an odd number of odd parts. G.f. : product(i=1,oo,1+(-1)^i*x^i) - Jon Perry (perry(AT)globalnet.co.uk), Jun 04 2004
Apart from signs, identical to A000009.
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FORMULA
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Euler transform of period 4 sequence [ -1, 1, -1, 0, ...].
Given g.f. A(x), B(x)=x* A(x^3)^8 satisfies 0=f(B(x), B(x^2)) where f(u, v)= (u-v^2)* (v-u^2) -(4*u*v* (1-u*v))^2.
G.f. is Fourier series of a weight 0 level 2304 modular form. f(-1/ (2304 t)) = f(t) where q = exp(2 pi i t). - Michael Somos Jul 16 2007
G.f.: Product_{k>0} 1/(1+x^(2k-1)).
Expansion of 1/chi(q) in powers of q where chi() is a Ramanujan theta function.
Expansion of q^(-1/24) * eta(q) * eta(q^4) / eta(q^2)^2 in powers of q.
Expansion of q^(-1/24) / f(t) in powers of q=exp(Pi i t) where f() is Weber's function.
G.f. is Fourier series of a weight 0 level 2304 modular form. f(-1/ (2304 t)) = f(t) where q = exp(2 pi i t).
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EXAMPLE
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q - q^25 + q^49 - 2*q^73 + 2*q^97 - 3*q^121 + 4*q^145 - 5*q^169 + ...
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MAPLE
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read theta; t1:=series(eta, q, 48); t2:= q^(-1/24)*t1*subs(q=q^4, t1)/subs(q=q^2, t1)^2; series(t2, q, 48); seriestolist(%); - N. J. A. Sloane (njas(AT)research.att.com), Aug 24 2007
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PROGRAM
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(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)/eta(x^2+A)^2, n))}
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CROSSREFS
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A000009(n)=(-1)^n a(n). Convolution inverse of A000700.
Sequence in context: A058703 A034320 A000009 this_sequence A117409 A092833 A100926
Adjacent sequences: A081357 A081358 A081359 this_sequence A081361 A081362 A081363
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Mar 18 2003
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