%I #26 Apr 25 2017 12:11:39

%S 1,1,21,2541,1023561,1036809081,2219782435101,8923051855107621,

%T 61797392100611962641,690766390156657904866161,

%U 11839493254591562294152214181,298556076626963858753929987732701,10706038142052878970311146962646277721,530588758323899225681861502684757146635241

%N Coefficients in the expansion of the sine lemniscate function.

%C Denoted \alpha_n by Lomont and Brillhart on page xii.

%D J. S. Lomont and J. Brillhart, Elliptic Polynomials, Chapman and Hall, 2001; see p. 87.

%H N. J. A. Sloane, <a href="/A144853/b144853.txt">Table of n, a(n) for n = 0..100</a>

%H P. Bala, <a href="/A144853/a144853.pdf">A triangle for calculating A144853</a>

%F 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

%F 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.

%F G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b = A187756. - _Michael Somos_, Jan 03 2013

%e G.f. = 1 + x + 21*x^2 + 2541*x^3 + 1023561*x^4 + 1036809081*x^5 + ...

%p 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);

%p a[n]:=(1/3)*add(binomial(4*n-1,4*j+1)*a[j]*b[n-1-j],j=0..n-1); od:

%p ta:=[seq(a[n],n=0..15)];

%t 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 *)

%t 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 *)

%o (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 */

%o (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 */

%Y Cf. A144849, A187756.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Feb 12 2009