A144849 Coefficients in the expansion of the squared sine lemniscate function

1, 6, 336, 77616, 50916096, 76307083776, 226653840838656, 1207012936807028736, 10696277678308486742016, 148900090457044541209706496, 3110043187741674836967136690176, 93885206124269301790338015801901056, 3970859549814416912519992571903015387136

Offset

0,2

Comments

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

Gives the number of Increasing bilabeled strict binary trees with 4n+2 labels. - Markus Kuba, Nov 18 2014

References

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

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..100

O. Bodini, M. Dien, X. Fontaine, A. Genitrini, and H. K. Hwang, Increasing Diamonds, 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.

Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], (17-November-2014).

Tanay Wakhare, Christophe Vignat, Taylor coefficients of the Jacobi theta3(q) function, arXiv:1909.01508 [math.NT], 2019.

Eric Weisstein's World of Mathematics, Lemniscate Constant

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).

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

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

Example

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

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)];

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 *)

Prog

(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 */

Crossrefs

Cf. A064853, A144853.

Sequence in context: A295925 A210769 A003031 * A212490 A047941 A229501

Adjacent sequences: A144846 A144847 A144848 * A144850 A144851 A144852

Keyword

nonn

Author

N. J. A. Sloane, Feb 12 2009

Status

approved