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A144849
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Coefficients in the expansion of the squared sine lemniscate function.
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7
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1, 6, 336, 77616, 50916096, 76307083776, 226653840838656, 1207012936807028736, 10696277678308486742016, 148900090457044541209706496, 3110043187741674836967136690176, 93885206124269301790338015801901056, 3970859549814416912519992571903015387136
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Denoted \beta_n by Lomont and Brillhart on page xiii.
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REFERENCES
| J. S. Lomont and J. Brillhart, Elliptic Polynomials, Chapman and Hall, 2001; see p. 86.
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..100
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FORMULA
| E.g.f.: sl(x)^2 = 2 Sum_{k>=0} (-12)^k * a(k) * x^(4*k + 2) / (4*k + 2)! where sl(x) = sin lemn(x) is the sine lemniscate function of Gauss. - Michael Somos, Apr 25 2011
a(0) = 1, a(n + 1) = Sum_{j=0..n} binomial( 4*n + 4, 4*j + 2) * a(j) * a(n - j).
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MAPLE
| a[0]:=1; b[0]:=1;
for n from 1 to 15 do b[n]:=add(binomial(4*n, 4*j+2)*b[j]*b[n-1-j], j=0..n-1);
a[n]:=(1/3)*add(binomial(4*n-1, 4*j+1)*a[j]*b[n-1-j], j=0..n-1); od:
tb:=[seq(b[n], n=0..15)];
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MATHEMATICA
| a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 2}, m! SeriesCoefficient[ JacobiSD[ x, 1/2]^2, {x, 0, m}] / (2 (-3)^n)]] (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 2}, m! SeriesCoefficient[ InverseSeries[ Integrate[ Series[(1 + x^4 / 12) ^ (-1/2), {x, 0, m + 1}], x]]^2 / 2, {x, 0, m}]]] (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[ n < 1, Boole[n == 0], Sum[ Binomial[ 4 n, 4 j + 2] a[j] a[ n - 1 - j], {j, 0, n - 1}]] (* Michael Somos, Apr 25 2011 *)
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PROG
| (PARI) {a(n) = local(m); if( n<0, 0, m = 4*n + 2; m! * polcoeff( (serreverse( intformal( (1 + x^4 / 12 + x * O(x^m)) ^ (-1/2))))^2 / 2 , m))} /* Michael Somos, Apr 25 2011 */
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CROSSREFS
| Cf. A144853.
Sequence in context: A135195 A001509 A003031 * A047941 A000409 A059415
Adjacent sequences: A144846 A144847 A144848 * A144850 A144851 A144852
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Feb 12 2009
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