A144853 Coefficients in the expansion of the sine lemniscate function.
1, 1, 21, 2541, 1023561, 1036809081, 2219782435101, 8923051855107621, 61797392100611962641, 690766390156657904866161, 11839493254591562294152214181, 298556076626963858753929987732701, 10706038142052878970311146962646277721, 530588758323899225681861502684757146635241
Offset: 0
Keywords
Examples
G.f. = 1 + x + 21*x^2 + 2541*x^3 + 1023561*x^4 + 1036809081*x^5 + ...
References
- J. S. Lomont and J. Brillhart, Elliptic Polynomials, Chapman and Hall, 2001; see p. 87.
Links
- N. J. A. Sloane, Table of n, a(n) for n = 0..100
- P. Bala, A triangle for calculating A144853
Programs
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Maple
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: ta:=[seq(a[n],n=0..15)];
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Mathematica
a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 1}, m! SeriesCoefficient[ JacobiSD[ x, 1/2], {x, 0, m}] / (-3)^n]]; (* Michael Somos, Apr 25 2011 *) a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 1}, m! SeriesCoefficient[ InverseSeries[ Integrate[ Series[ (1 + x^4 / 12) ^ (-1/2), {x, 0, m + 1}], x]], {x, 0, m}]]]; (* Michael Somos, Apr 25 2011 *) -
PARI
{a(n) = my(m); if( n<0, 0, m = 4*n + 1; m! * polcoeff( serreverse( intformal( (1 + x^4 / 12 + x * O(x^m)) ^ (-1/2))), m))}; /* Michael Somos, Apr 25 2011 */ -
PARI
{a(n) = my(A, m); if( n<0, 0, m = 4*n + 1; A = O(x); for( k=0, n, A = x + intformal( intformal( A^3 / 6))); m! * polcoeff( A, m))}; /* Michael Somos, Apr 25 2011 */
Formula
E.g.f.: sl(x) = Sum_{k>=0} (-12)^k * a(k) * x^(4*k + 1) / (4*k + 1)! where sl(x) = sin lemn(x) is the sine lemniscate function of Gauss. - Michael Somos, Apr 25 2011
a(0) = 1, a(n + 1) = (1 / 3) * Sum_{j=0..n} binomial( 4*n + 3, 4*j + 1) * a(j) * b(n - j) where b() is A144849.
G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b = A187756. - Michael Somos, Jan 03 2013
Comments