cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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.

Original entry on oeis.org

1, 1, 9, 153, 4977, 261009, 20039481, 2121958377, 296297348193, 52750142341281, 11662264481073129, 3134732109393169593, 1006734732695870345937, 380718482718134681818929, 167456229155543640166939161, 84761007600911799530893148937, 48919649166315485705652984573633
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2011

Keywords

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.

Examples

			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-...))))))))).

Links

Crossrefs

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

Programs

  • 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 *)
  • 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

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

A193544 E.g.f.: sqrt(2)*(L/Pi) / (1 + 2*Sum_{n>=1} cosh(2*Pi*n*x/L)/cosh(n*Pi)) where L = Lemniscate constant.

Original entry on oeis.org

1, -1, -3, 27, 441, -11529, -442827, 23444883, 1636819569, -145703137041, -16106380394643, 2164638920874507, 347592265948756521, -65724760945840254489, -14454276753061349098587, 3658147171522531111996803, 1055646229815910768764248289
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2011

Keywords

Comments

L = Lemniscate constant = 2*(Pi/2)^(3/2)/gamma(3/4)^2 = 2.62205755429...
Compare the definition with that of the dual sequence A193541.

Examples

			E.g.f.: A(x) = 1 - x^2/2! - 3*x^4/4! + 27*x^6/6! + 441*x^8/8! - 11529*x^10/10! - 442827*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 A193541, 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! - 3*x^4/4! - 27*x^6/6! + 441*x^8/8! + 11529*x^10/10! - 442827*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 - 3*x^2 + 27*x^3 + 441*x^4 - 11529*x^5 - 442827*x^6 +...+ a(n)*x^n +...
O.g.f.: 1/(1 + x/(1 - 4*x/(1 + 9*x/(1 - 16*x/(1 + 25*x/(1 - 36*x/(1 + 49*x/(1 - 64*x/(1+...))))))))).

Links

Crossrefs

Cf. A159600, A193540, A193541, A193542, A193543, A193545.

Programs

  • Mathematica
    L = 2*(Pi/2)^(3/2)/Gamma[3/4]^2; a[0] = 1; a[n_] := 2*Pi/L*NSum[(-1)^k * (2*k*Pi/L)^(2*n)/Cosh[k*Pi], {k, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 50] // Round; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Sep 29 2017 *)
  • PARI
    {a(n)=local(L=2*(Pi/2)^(3/2)/gamma(3/4)^2); if(n==0, 1, 2*Pi/L*suminf(k=1, (-1)^k*(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=(sqrt(2)*L/Pi)/(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/L)*(1 + 2*suminf(m=1,(-1)^m/(1 - (2*m*Pi/L)^2*x+x*O(x^n))/cosh(m*Pi)));
round(polcoeff(R,n))} \\ Paul D. Hanna, Aug 29 2012

Formula

Given e.g.f. A(x), define the e.g.f. B(x) of A193541:
B(x) = sqrt(2)*L / (Pi*(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 A193543.
...
O.g.f.: 1/(1 + 1^2*x/(1 - 2^2*x/(1 + 3^2*x/(1 - 4^2*x/(1 + 5^2*x/(1 - 6^2*x/(1 + 7^2*x/(1 - 8^2*x/(1+...))))))))) (continued fraction).
O.g.f.: (Pi/L) * (1 + 2*Sum_{n>=1} (-1)^n/(1 - (2*n*Pi/L)^2*x) / cosh(n*Pi)) where L = Lemniscate constant. - Paul D. Hanna, Aug 29 2012
...
a(n) = 2*Pi/L * Sum_{k>=1} (-1)^k*(2*k*Pi/L)^(2*n) / cosh(k*Pi) for n>0 where L = Lemniscate constant. - Paul D. Hanna, Aug 29 2012
G.f.: 1/Q(0), where Q(k)= 1 + x*(2*k+1)^2/(1 - x*(2*k+2)^2/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 27 2013
G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)^2/(x*(2*k+1)^2 + 1/(1 - x*(2*k+2)^2/(x*(2*k+2)^2 - 1/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013

A193541 E.g.f.: sqrt(2)*L / (Pi*(1 + 2*Sum_{n>=1} cos(2*Pi*n*x/L)/cosh(n*Pi) )) where L = Lemniscate constant.

Original entry on oeis.org

1, 1, -3, -27, 441, 11529, -442827, -23444883, 1636819569, 145703137041, -16106380394643, -2164638920874507, 347592265948756521, 65724760945840254489, -14454276753061349098587, -3658147171522531111996803, 1055646229815910768764248289
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2011

Keywords

Comments

L = Lemniscate constant = 2*(Pi/2)^(3/2)/gamma(3/4)^2 = 2.62205755429...
Compare the definition with that of the dual sequence A193544.

Examples

			E.g.f.: A(x) = 1 + x^2/2! - 3*x^4/4! - 27*x^6/6! + 441*x^8/8! + 11529*x^10/10! - 442827*x^12/12! +...+ a(n)*x^(2*n)/(2*n)! +...
where
A(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) +...)).
Let B(x) equal the e.g.f. of A193544, where:
B(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) +...))
explicitly,
B(x) = 1 - x^2/2! - 3*x^4/4! + 27*x^6/6! + 441*x^8/8! - 11529*x^10/10! - 442827*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 - 3*x^2 - 27*x^3 + 441*x^4 + 11529*x^5 - 442827*x^6 +...+ a(n)*x^n +...
O.g.f.: 1/(1 - x/(1 + 4*x/(1 - 9*x/(1 + 16*x/(1 - 25*x/(1 + 36*x/(1 - 49*x/(1 + 64*x/(1-...))))))))).

Links

Crossrefs

Cf. A159600, A193540, A193542, A193543, A193544, A193545.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[{m = 2 n}, 2^n m! SeriesCoefficient[ JacobiND[ x, 1/2], {x, 0, m}]]]; (* Michael Somos, Oct 18 2011 *)
a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ JacobiDN[ x, -1], {x, 0, m}]]]; (* Michael Somos, Jun 17 2016 *)
  • PARI
    {a(n)=local(R,L=2*(Pi/2)^(3/2)/gamma(3/4)^2);
R=(sqrt(2)*L/Pi)/(1 + 2*suminf(m=1,cos(2*Pi*m*x/L +O(x^(2*n+1)))/cosh(m*Pi)));
round((2*n)!*polcoeff(R,2*n))}

Formula

Given e.g.f. A(x), define the e.g.f. B(x) of A193544:
B(x) = sqrt(2)*L / (Pi*(1 + 2*Sum_{n>=1} cosh(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 A193540.
O.g.f.: 1/(1 - 1^2*x/(1 + 2^2*x/(1 - 3^2*x/(1 + 4^2*x/(1 - 5^2*x/(1 + 6^2*x/(1 - 7^2*x/(1 + 8^2*x/(1-...))))))))) (continued fraction).
G.f.: 1/U(0) where U(k)= 1 - x*(2*k+1)^2/(1 + x*(2*k+2)^2/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Jun 28 2012
G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)^2/(x*(2*k+1)^2 - 1/(1 - x*(2*k+2)^2/(x*(2*k+2)^2 + 1/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013

A193542 E.g.f.: 2*L^2/(Pi^2*(1 + 2*Sum_{n>=1} cos(2*Pi*n*x/L)/cosh(n*Pi) )^2) where L = Lemniscate constant.

Original entry on oeis.org

1, 0, 2, 0, 0, 0, -144, 0, 0, 0, 96768, 0, 0, 0, -268240896, 0, 0, 0, 2111592333312, 0, 0, 0, -37975288540299264, 0, 0, 0, 1353569484565546795008, 0, 0, 0, -86498911610371173437669376, 0, 0, 0, 9198407234012051081051108278272, 0, 0, 0, -1536583522302562247445395779495133184
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2011

Keywords

Comments

L = Lemniscate constant = 2*(Pi/2)^(3/2)/gamma(3/4)^2 = 2.62205755429...
Compare the definition with that of the dual sequence A193545.

Examples

			E.g.f.: A(x) = 1 + 2*x^2/2! - 144*x^6/6! + 96768*x^10/10! - 268240896*x^14/14! +...+ a(n)*x^n/n! +...
which equals the square of the e.g.f. B(x) of A193541:
B(x) = 1 + x^2/2! - 3*x^4/4! - 27*x^6/6! + 441*x^8/8! + 11529*x^10/10! - 442827*x^12/12! +...

Links

Crossrefs

Cf. A193540, A193541, A193543, A193544, A193545.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiDN[ x, -1]^2, {x, 0, n}]]; (* Michael Somos, Jun 17 2016 *)
  • PARI
    {a(n)=local(R,L=2*(Pi/2)^(3/2)/gamma(3/4)^2);
R=(sqrt(2)*L/Pi)/(1 + 2*suminf(m=1,cos(2*Pi*m*x/L +x*O(x^n))/cosh(m*Pi)));
round(n!*polcoeff(R^2,n))}

Formula

a(n) = -A193545(n) for n>=1.
E.g.f.: dn(x, -1)^2 where dn() is a Jacobi elliptic function. - Michael Somos, Jun 17 2016

A193545 E.g.f.: 2*L^2/(Pi^2*(1 + 2*Sum_{n>=1} cosh(2*Pi*n*x/L)/cosh(n*Pi) )^2) where L = Lemniscate constant.

Original entry on oeis.org

1, 0, -2, 0, 0, 0, 144, 0, 0, 0, -96768, 0, 0, 0, 268240896, 0, 0, 0, -2111592333312, 0, 0, 0, 37975288540299264, 0, 0, 0, -1353569484565546795008, 0, 0, 0, 86498911610371173437669376, 0, 0, 0, -9198407234012051081051108278272, 0, 0, 0, 1536583522302562247445395779495133184
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2011

Keywords

Comments

L = Lemniscate constant = 2*(Pi/2)^(3/2)/gamma(3/4)^2 = 2.62205755429...
Compare the definition with that of the dual sequence A193542.

Examples

			E.g.f.: A(x) = 1 - 2*x^2/2! + 144*x^6/6! - 96768*x^10/10! + 268240896*x^14/14! +...+ a(n)*x^n/n! +...
which equals the square of the e.g.f. B(x) of A193544:
B(x) = 1 - x^2/2! - 3*x^4/4! + 27*x^6/6! + 441*x^8/8! - 11529*x^10/10! - 442827*x^12/12! +...

Links

Crossrefs

Cf. A193540, A193541, A193542, A193543, A193544.

Programs

  • PARI
    {a(n)=local(R,L=2*(Pi/2)^(3/2)/gamma(3/4)^2);
R=(sqrt(2)*L/Pi)/(1 + 2*suminf(m=1,cosh(2*Pi*m*x/L +x*O(x^n))/cosh(m*Pi)));
round(n!*polcoeff(R^2,n))}

Formula

a(n) = -A193542(n) for n>=1.
Showing 1-5 of 5 results.

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