[go: up one dir, main page]

login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Coefficients in the expansion of the squared sine lemniscate function
4

%I #53 Jul 10 2024 17:26:12

%S 1,6,336,77616,50916096,76307083776,226653840838656,

%T 1207012936807028736,10696277678308486742016,

%U 148900090457044541209706496,3110043187741674836967136690176,93885206124269301790338015801901056,3970859549814416912519992571903015387136

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

%C Denoted by \beta_n in Lomont and Brillhart (2011) on page xiii.

%C Gives the number of Increasing bilabeled strict binary trees with 4n+2 labels. - _Markus Kuba_, Nov 18 2014

%D J. S. Lomont and J. Brillhart, Elliptic Polynomials, Chapman and Hall, 2001; see p. 86.

%H N. J. A. Sloane, <a href="/A144849/b144849.txt">Table of n, a(n) for n = 0..100</a>

%H 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>

%F 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 a(0) = 1, a(n + 1) = Sum_{j=0..n} binomial( 4*n + 4, 4*j + 2) * a(j) * a(n - j).

%F 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 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 G.f. = 1 + 6*x + 336*x^2 + 77616*x^3 + 50916096*x^4 + ...

%p a[0]:=1; b[0]:=1;

%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 tb:=[seq(b[n],n=0..15)];

%t 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 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 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 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 (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 Cf. A064853, A144853.

%K nonn

%O 0,2

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