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A022577
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Expansion of prod( m>=1, 1 + x^m )^12.
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6
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1, 12, 78, 376, 1509, 5316, 16966, 50088, 138738, 364284, 913824, 2203368, 5130999, 11585208, 25444278, 54504160, 114133296, 234091152, 471062830, 931388232, 1811754522, 3471186596, 6556994502, 12222818640, 22502406793, 40944396120
(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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f.: prod(k>=1, 1 + x^k )^12.
Expansion of chi(-x)^-12 in powers of x where chi() is a Ramanujan theta function.
Expansion of k^2 / (16 * q * k') in powers of q^2. - Michael Somos, Jul 22 2011
Expansion of q^(-1/2) * (k/4) / (1 - k^2) in powers of q. - Michael Somos, Jul 16 2005
Expansion of q^(-1/2) * (eta(q^2) / eta(q))^12 in powers of q. - Michael Somos, Jul 16 2005
Euler transform of period 2 sequence [12, 0, ...]. - Michael Somos, Jul 16 2005
Given g.f. A(x), then B(x) = (x * A(x^2))^2 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (4096*u*v + 48*u + 1)*v - u^2 . - Michael Somos, Jul 16 2005
G.f. is a period 1 Fourier series which satisfies f( -1/(8*t)) = 1 / (64 * f(t)) where q = exp(2*pi*i*t). - Michael Somos, Jul 22 2011
A124863(n) = (-1)^n * a(n). A007096(4*n + 2) = 8 * a(n). Convolution inverse of A007249.
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EXAMPLE
| 1 + 12*x + 78*x^2 + 376*x^3 + 1509*x^4 + 5316*x^5 + 16966*x^6 + ...
q + 12*q^3 + 78*q^5 + 376*q^7 + 1509*q^9 + 5316*q^11 + 16966*q^13 + ...
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MATHEMATICA
| a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q)^(1/2) / (1 - m), {q, 0, n}]] (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q) / (1 - m)^(1/2), {q, 0, 2 n}]] (* Michael Somos, Jul 22 2011 *)
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PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 + x^k, 1 + x * O(x^n))^12, n))} /* Michael Somos, Jul 16 2005 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^12, n))} /* Michael Somos, Jul 16 2005 */
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CROSSREFS
| Cf. A007096, A007249, A124863.
Sequence in context: A001288 A121665 A124863 * A189493 A199492 A200055
Adjacent sequences: A022574 A022575 A022576 * A022578 A022579 A022580
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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