A193543 E.g.f.: Pi/(sqrt(2)*L) * (1 + 2*Sum_{n>=1} cosh(2*Pi*n*x/L)/cosh(n*Pi)) where L = Lemniscate constant.

1, 1, 9, 153, 4977, 261009, 20039481, 2121958377, 296297348193, 52750142341281, 11662264481073129, 3134732109393169593, 1006734732695870345937, 380718482718134681818929, 167456229155543640166939161, 84761007600911799530893148937, 48919649166315485705652984573633

Offset

0,3

Comments

L = Lemniscate constant = 2*(Pi/2)^(3/2)/gamma(3/4)^2 = 2.62205755429...

Compare the definition with that of the dual sequence A193540.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..234

Eric Weisstein's World of Mathematics, Ramanujan Cos/Cosh Identity.

Formula

E.g.f.: cosh( Series_Reversion( Integral 1/sqrt( cosh(2*x) ) dx ) ). - Paul D. Hanna, Aug 14 2017

E.g.f.: sqrt(1 + S(x)^2), where S(x) is the e.g.f. of A289695. - Paul D. Hanna, Aug 14 2017

E.g.f.: 1 + Integral S(x) * sqrt(1 + 2*S(x)^2) dx, where S(x) is the e.g.f. of A289695. - Paul D. Hanna, Aug 14 2017

...

Given e.g.f. A(x), define the e.g.f. of A193540:

B(x) = Pi/(sqrt(2)*L) * (1 + 2*Sum_{n>=1} cos(2*Pi*n*x/L) / cosh(n*Pi)),

then A(x)^-2 + B(x)^-2 = 2 by Ramanujan's cos/cosh identity.

...

E.g.f. equals the reciprocal of the e.g.f. of A193544.

...

O.g.f.: 1/(1 - 1^2*x/(1 - 2*2^2*x/(1 - 3^2*x/(1 - 2*4^2*x/(1 - 5^2*x/(1 - 2*6^2*x/(1 - 7^2*x/(1 - 2*8^2*x/(1-...))))))))) (continued fraction).

O.g.f.: Pi/(sqrt(2)*L) * (1 + 2*Sum_{n>=1} 1/(1 - (2*n*Pi/L)^2*x) / cosh(n*Pi)) where L = Lemniscate constant. - Paul D. Hanna, Aug 29 2012

...

a(n) = sqrt(2)*Pi/L * Sum_{k>=1} (2*k*Pi/L)^(2*n) / cosh(k*Pi) for n>0 where L = Lemniscate constant. - Paul D. Hanna, Aug 29 2012

...

G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)^2/(x*(2*k+1)^2 - 1/(1 - 2*x*(2*k+2)^2/(2*x*(2*k+2)^2 - 1/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013

a(n) ~ 2^(7*n + 4) * Pi^(n+1) * n^(2*n + 1/2) / (exp(2*n) * Gamma(1/4)^(4*n + 2)). - Vaclav Kotesovec, Nov 29 2020

Example

E.g.f.: A(x) = 1 + x^2/2! + 9*x^4/4! + 153*x^6/6! + 4977*x^8/8! + 261009*x^10/10! + 20039481*x^12/12! +...+ a(n)*x^(2*n)/(2*n)! +...
where
A(x)*sqrt(2)*L/Pi = 1 + 2*cosh(2*Pi*x/L)/cosh(Pi) + 2*cosh(4*Pi*x/L)/cosh(2*Pi) + 2*cosh(6*Pi*x/L)/cosh(3*Pi) +...
Let B(x) equal the e.g.f. of A193540, where:
B(x)*sqrt(2)*L/Pi = 1 + 2*cos(2*Pi*x/L)/cosh(Pi) + 2*cos(4*Pi*x/L)/cosh(2*Pi) + 2*cos(6*Pi*x/L)/cosh(3*Pi) +...
explicitly,
B(x) = 1 - x^2/2! + 9*x^4/4! - 153*x^6/6! + 4977*x^8/8! - 261009*x^10/10! + 20039481*x^12/12! +...
then A(x)^-2 + B(x)^-2 = 2
as illustrated by:
A(x)^-2 = 1 - 2*x^2/2! + 144*x^6/6! - 96768*x^10/10! + 268240896*x^14/14! +...
B(x)^-2 = 1 + 2*x^2/2! - 144*x^6/6! + 96768*x^10/10! - 268240896*x^14/14! +...
...
O.g.f.: 1 + x + 9*x^2 + 153*x^3 + 4977*x^4 + 261009*x^5 + 20039481*x^6 +...+ a(n)*x^n +...
O.g.f.: 1/(1 - x/(1 - 8*x/(1 - 9*x/(1 - 32*x/(1 - 25*x/(1 - 72*x/(1 - 49*x/(1 - 128*x/(1-...))))))))).

Mathematica

nmax = 20; s = CoefficientList[Series[JacobiDN[Sqrt[2]*x, 1/2], {x, 0, 2*nmax}], x] * Range[ 0, 2*nmax]!; Table[(-1)^n * s[[2*n + 1]], {n, 0, nmax}] (* Vaclav Kotesovec, Nov 29 2020 *)

Prog

(PARI) {a(n)=local(L=2*(Pi/2)^(3/2)/gamma(3/4)^2); if(n==0, 1, sqrt(2)*Pi/L*suminf(k=1, (2*k*Pi/L)^(2*n)/cosh(k*Pi)))} \\ Paul D. Hanna, Aug 29 2012
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(R, L=2*(Pi/2)^(3/2)/gamma(3/4)^2);
R=Pi/(sqrt(2)*L)*(1 + 2*suminf(m=1, cosh(2*Pi*m*x/L +O(x^(2*n+1)))/cosh(m*Pi)));
round((2*n)!*polcoeff(R, 2*n))}
(PARI) {a(n)=local(R, L=2*(Pi/2)^(3/2)/gamma(3/4)^2);
R=Pi/(sqrt(2)*L)*(1 + 2*suminf(m=1, 1/(1 - (2*m*Pi/L)^2*x+x*O(x^n))/cosh(m*Pi)));
round(polcoeff(R, n))} \\ Paul D. Hanna, Aug 29 2012
(PARI) {a(n) = my(C=1); C = cosh( serreverse( intformal( 1/sqrt( cosh(2*x +O(x^(2*n+1))) ) ) ) ); (2*n)!*polcoeff(C, 2*n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 14 2017

Crossrefs

Cf. A193540, A193541, A193542, A193544, A193545.

Sequence in context: A009037 A012148 A193540 * A173982 A185759 A215557

Adjacent sequences: A193540 A193541 A193542 * A193544 A193545 A193546

Keyword

nonn

Author

Paul D. Hanna, Jul 29 2011

Status

approved