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A144849 Coefficients in the expansion of the squared sine lemniscate function.

%I A144849 #56 Jul 22 2025 06:12:00
%S A144849 1,6,336,77616,50916096,76307083776,226653840838656,
%T A144849 1207012936807028736,10696277678308486742016,
%U A144849 148900090457044541209706496,3110043187741674836967136690176,93885206124269301790338015801901056,3970859549814416912519992571903015387136
%N A144849 Coefficients in the expansion of the squared sine lemniscate function.
%C A144849 Denoted by \beta_n in Lomont and Brillhart (2011) on page xiii.
%C A144849 Gives the number of Increasing bilabeled strict binary trees with 4n+2 labels. - _Markus Kuba_, Nov 18 2014
%D A144849 J. S. Lomont and J. Brillhart, Elliptic Polynomials, Chapman and Hall, 2001; see p. 86.
%H A144849 N. J. A. Sloane, <a href="/A144849/b144849.txt">Table of n, a(n) for n = 0..100</a>
%H A144849 O. Bodini, M. Dien, X. Fontaine, A. Genitrini, and H. K. Hwang, <a href="https://doi.org/10.1007/978-3-662-49529-2_16">Increasing Diamonds</a>, in LATIN 2016: 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings Pages pp 207-219 2016 DOI 10.1007/978-3-662-49529-2_16; Lecture Notes in Computer Science Series Volume 9644.
%H A144849 Markus Kuba, Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], (17-November-2014).
%H A144849 Tanay Wakhare, Christophe Vignat, <a href="https://arxiv.org/abs/1909.01508">Taylor coefficients of the Jacobi theta3(q) function</a>, arXiv:1909.01508 [math.NT], 2019.
%H A144849 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>
%F A144849 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
%F A144849 a(0) = 1, a(n + 1) = Sum_{j=0..n} binomial( 4*n + 4, 4*j + 2) * a(j) * a(n - j).
%F A144849 G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b(n) = A139757(n) * n/3. - _Michael Somos_, Jan 03 2013
%F A144849 E.g.f.: Increasing bilabeled strict binary trees of 2n+2 labels (including the zeros): T(z)=Sum_{n>=1}T_n z^{2n}/(2n)! = 6/sqrt(3)*WeierstrassP(3^{-1/4}z+LemniscateConstant; g_2,g_3), with g_2=-1 and g_3=0; alternatively, T(z)=sqrt(3)*i*sl^2(z/(3^{1/4}(1+i))). - _Markus Kuba_, Nov 18 2014
%e A144849 G.f. = 1 + 6*x + 336*x^2 + 77616*x^3 + 50916096*x^4 + ...
%p A144849 a[0]:=1; b[0]:=1;
%p A144849 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 A144849 a[n]:=(1/3)*add(binomial(4*n-1,4*j+1)*a[j]*b[n-1-j],j=0..n-1); od:
%p A144849 tb:=[seq(b[n],n=0..15)];
%t A144849 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 *)
%t A144849 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 *)
%t A144849 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 *)
%t A144849 a[ n_] := If[n < 0, 0, With[{m = 4*n + 2}, m!*SeriesCoefficient[JacobiSN[x, -1]^2, {x, 0, m}]/(2*(-12)^n)]]; (* _Michael Somos_, Jul 10 2024 *)
%o A144849 (PARI) {a(n) = my(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 */
%Y A144849 Cf. A064853, A144853.
%K A144849 nonn
%O A144849 0,2
%A A144849 _N. J. A. Sloane_, Feb 12 2009