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 1-9 of 9 results.

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

A193540 E.g.f.: Pi/(sqrt(2)*L) * (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, 9, -153, 4977, -261009, 20039481, -2121958377, 296297348193, -52750142341281, 11662264481073129, -3134732109393169593, 1006734732695870345937, -380718482718134681818929, 167456229155543640166939161, -84761007600911799530893148937
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 A193543.

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*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 A193543, 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! + 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. A193541, A193542, A193543, A193544, A193545.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ Tan[ JacobiAmplitude[ x, -1]] / Tan[ JacobiAmplitude[ 2 x, -1] / 2], {x, 0, m}]]]; (* Michael Somos, Oct 18 2011 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ JacobiND[ x, -1], {x, 0, m}]]]; (* Michael Somos, Oct 18 2011 *)
Table[SeriesCoefficient[InverseSeries[Series[EllipticF[x, 1/2], {x, 0, 32}]], 2 n + 1] (2 n + 1)! 2^n, {n, 0, 15}] (* Benedict W. J. Irwin, Apr 04 2017 *)
Table[SeriesCoefficient[JacobiDN[Sqrt[2] x, 1/2], {x, 0, 2 k}] (2 k)!, {k, 0, 20}] (* Jan Mangaldan, Nov 28 2020 *)
nmax = 20; s = CoefficientList[Series[JacobiDN[Sqrt[2] x, 1/2], {x, 0, 2*nmax}], x] * Range[ 0, 2*nmax]!; Table[s[[2*n + 1]], {n, 0, nmax}] (* Vaclav Kotesovec, Nov 29 2020 *)
  • 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,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. of A193543:
B(x) = sqrt(2)*Pi/(2*L) * (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 A193541.
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).
G.f.: 1/Q(0) where p=2, Q(k) = 1 + x*(2*k+1)^2/( 1 + p*x*(2*k+2)^2/Q(k+1) ); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Mar 22 2013
a(n) ~ (-1)^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

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.

A289695 E.g.f.: S(x) satisfies: S(x) = Integral sqrt( (1 + S(x)^2)*(1 + 2*S(x)^2) ) dx.

Original entry on oeis.org

1, 3, 33, 819, 34209, 2189187, 198717057, 24278289651, 3842052205761, 764478393601923, 186805627856569953, 54994092004290217779, 19197418751181422089569, 7840711973025043515377667, 3704137338316764145483007937, 2004220869541285849551954747891
Offset: 1

Views

Author

Paul D. Hanna, Aug 14 2017

Keywords

Examples

			E.g.f.: S(x) = x + 3*x^3/3! + 33*x^5/5! + 819*x^7/7! + 34209*x^9/9! + 2189187*x^11/11! + 198717057*x^13/13! + 24278289651*x^15/15! + 3842052205761*x^17/17! + 764478393601923*x^19/19! + 186805627856569953*x^21/21! +...
where
sqrt((1 + S(x)^2)*(1 + 2*S(x)^2)) = 1 + 3*x^2/2! + 33*x^4/4! + 819*x^6/6! + 34209*x^8/8! +...
RELATED SERIES.
C(x) = sqrt(1 + S(x)^2) equals the e.g.f. of A193543, and begins
C(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! + 2121958377*x^14/14! + 296297348193*x^16/16! +...

Links

Crossrefs

Cf. A193543.

Programs

  • Mathematica
    a[ n_] := If[n<1, 0, (2*n-1)!*SeriesCoefficient[JacobiSN[I*x, 2]/I, {x, 0, 2*n-1}]]; (* Michael Somos, Feb 13 2025 *)
  • PARI
    {a(n) = my(S=1); S = sinh( serreverse( intformal( 1/sqrt( cosh(2*x +O(x^(2*n+1))) ) ) ) ); (2*n-1)!*polcoeff(S,2*n-1)}
for(n=1,20,print1(a(n),", "))

Formula

E.g.f.: S(x) = Series_Reversion( Integral 1/sqrt( (1 + x^2)*(1 + 2*x^2) ) dx ).
E.g.f.: S(x) = sinh( Series_Reversion( Integral 1/sqrt( cosh(2*x) ) dx ) ).
E.g.f.: S(x) = Integral sqrt(1 + S(i*x)^2) / (1 + 2*S(i*x)^2) dx, where i^2 = -1.
Let C(x) be the e.g.f. of A193543, then
(1) S(x) = sqrt(C(x)^2 - 1),
(2) S(x) = Integral C(x) * sqrt(C(x)^2 + S(x)^2) dx,
(3) C(x) = 1 + Integral S(x) * sqrt(C(x)^2 + S(x)^2) dx,
(4) C(x) + S(x) = exp( Integral sqrt(1 + 2*S(x)^2) dx ).

A292181 E.g.f. A(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that A'(x) = A(x) + B(x)*C(x).

Original entry on oeis.org

1, 3, 10, 45, 259, 1806, 14827, 140367, 1504576, 17972559, 236275711, 3387012720, 52572376669, 878552787927, 15729439074058, 300400031036745, 6095885898471775, 130982551821899862, 2970844882925223487, 70929401617621416243, 1778125633605205346584, 46698342082602696345555, 1282168260097348871508667, 36734284970419645262875200, 1096293296048734274708523433, 34026339905854090378353208155
Offset: 1

Views

Author

Paul D. Hanna, Sep 10 2017

Keywords

Comments

Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.
Another Pythagorean triple is the e.g.f.s of A289695, A193543, and A153302, which are related to the Lemniscate sine and cosine functions, sl(x) and cl(x).

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 10*x^3/3! + 45*x^4/4! + 259*x^5/5! + 1806*x^6/6! + 14827*x^7/7! + 140367*x^8/8! + 1504576*x^9/9! + 17972559*x^10/10! + 236275711*x^11/11! + 3387012720*x^12/12! + 52572376669*x^13/13! + 878552787927*x^14/14! + 15729439074058*x^15/15! + 300400031036745*x^16/16! +...
where A(x) = Integral A(x) + B(x)*C(x) dx.
RELATED SERIES.
B(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 35*x^4/4! + 226*x^5/5! + 1715*x^6/6! + 14701*x^7/7! + 141248*x^8/8! + 1515661*x^9/9! + 18048527*x^10/10! + 236581984*x^11/11! + 3386091821*x^12/12! + 52533799501*x^13/13! + 877993866290*x^14/14! + 15723411375931*x^15/15! + 300349139257727*x^16/16 +...
where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
C(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 63*x^4/4! + 361*x^5/5! + 2499*x^6/6! + 20581*x^7/7! + 196311*x^8/8! + 2116561*x^9/9! + 25357563*x^10/10! + 333765037*x^11/11! + 4787007855*x^12/12! + 74323701817*x^13/13! + 1242253733619*x^14/14! + 22243082373301*x^15/15! + 424815246293319*x^16/16! +...
where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
Squares of series.
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 134*x^4/4! + 1050*x^5/5! + 9158*x^6/6! + 89418*x^7/7! + 972470*x^8/8! + 11700378*x^9/9! + 154613222*x^10/10! + 2227684074*x^11/11! + 34757852054*x^12/12! + 583740365754*x^13/13! + 10497898450118*x^14/14! + 201267889853706*x^15/15! + 4097952119101814*x^16/16! +...
where A(x)^2 + B(x)^2 = C(x)^2.
B(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 150*x^4/4! + 1082*x^5/5! + 9222*x^6/6! + 89546*x^7/7! + 972726*x^8/8! + 11700890*x^9/9! + 154614246*x^10/10! + 2227686122*x^11/11! + 34757856150*x^12/12! + 583740373946*x^13/13! + 10497898466502*x^14/14! + 201267889886474*x^15/15! + 4097952119167350*x^16/16! +...
where B(x)^2 - A(x)^2 = exp(2*x).
C(x)^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 284*x^4/4! + 2132*x^5/5! + 18380*x^6/6! + 178964*x^7/7! + 1945196*x^8/8! + 23401268*x^9/9! + 309227468*x^10/10! + 4455370196*x^11/11! + 69515708204*x^12/12! + 1167480739700*x^13/13! + 20995796916620*x^14/14! + 402535779740180*x^15/15! + 8195904238269164*x^16/16! +...
where C(x)^2 - 2*A(x)^2 = exp(2*x).

Links

Crossrefs

Cf. A292182 (B), A292183 (C).

Programs

  • PARI
    {a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f. A(x) and related functions B(x) and C(x) satisfy:
(1a) A(x)^2 + B(x)^2 = C(x)^2.
(1b) B(x)^2 - A(x)^2 = exp(x)^2.
(1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
(2a) A(x) = Integral A(x) + B(x)*C(x) dx.
(2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
(2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
(3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
(3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
(3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
(3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
(4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
(4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
(4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
(5a) B(x) + i*A(x) = C(x) * exp( i*atan( A(x)/B(x) ) ).
(5b) A(x)/B(x) = Series_Reversion( Integral 1/( sqrt(1-x^4) * (1 + Integral 1/sqrt(1-x^4) dx) ) dx ).
Limit A292182(n)/A292181(n) = 1.
Limit A292183(n)/A292181(n) = sqrt(2).

A292182 E.g.f. B(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that B'(x) = B(x) + A(x)*C(x).

Original entry on oeis.org

1, 1, 2, 7, 35, 226, 1715, 14701, 141248, 1515661, 18048527, 236581984, 3386091821, 52533799501, 877993866290, 15723411375931, 300349139257727, 6095613429234730, 130983518612114231, 2970900143887175977, 70930381205350706888, 1778137090832694851161, 46698407537794612100459, 1282167191852237842607584, 36734238381564939631425737, 1096292258727541156091352361, 34026322932421876848090674594
Offset: 0

Views

Author

Paul D. Hanna, Sep 10 2017

Keywords

Comments

Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.
Another Pythagorean triple is the e.g.f.s of A289695, A193543, and A153302, which are related to the Lemniscate sine and cosine functions, sl(x) and cl(x).

Examples

			E.g.f.: B(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 35*x^4/4! + 226*x^5/5! + 1715*x^6/6! + 14701*x^7/7! + 141248*x^8/8! + 1515661*x^9/9! + 18048527*x^10/10! + 236581984*x^11/11! + 3386091821*x^12/12! + 52533799501*x^13/13! + 877993866290*x^14/14! + 15723411375931*x^15/15! + 300349139257727*x^16/16 +...
where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
RELATED SERIES.
A(x) = x + 3*x^2/2! + 10*x^3/3! + 45*x^4/4! + 259*x^5/5! + 1806*x^6/6! + 14827*x^7/7! + 140367*x^8/8! + 1504576*x^9/9! + 17972559*x^10/10! + 236275711*x^11/11! + 3387012720*x^12/12! + 52572376669*x^13/13! + 878552787927*x^14/14! + 15729439074058*x^15/15! + 300400031036745*x^16/16! +...
where A(x) = Integral A(x) + B(x)*C(x) dx.
C(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 63*x^4/4! + 361*x^5/5! + 2499*x^6/6! + 20581*x^7/7! + 196311*x^8/8! + 2116561*x^9/9! + 25357563*x^10/10! + 333765037*x^11/11! + 4787007855*x^12/12! + 74323701817*x^13/13! + 1242253733619*x^14/14! + 22243082373301*x^15/15! + 424815246293319*x^16/16! +...
where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
Squares of series.
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 134*x^4/4! + 1050*x^5/5! + 9158*x^6/6! + 89418*x^7/7! + 972470*x^8/8! + 11700378*x^9/9! + 154613222*x^10/10! + 2227684074*x^11/11! + 34757852054*x^12/12! + 583740365754*x^13/13! + 10497898450118*x^14/14! + 201267889853706*x^15/15! + 4097952119101814*x^16/16! +...
where A(x)^2 + B(x)^2 = C(x)^2.
B(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 150*x^4/4! + 1082*x^5/5! + 9222*x^6/6! + 89546*x^7/7! + 972726*x^8/8! + 11700890*x^9/9! + 154614246*x^10/10! + 2227686122*x^11/11! + 34757856150*x^12/12! + 583740373946*x^13/13! + 10497898466502*x^14/14! + 201267889886474*x^15/15! + 4097952119167350*x^16/16! +...
where B(x)^2 - A(x)^2 = exp(2*x).
C(x)^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 284*x^4/4! + 2132*x^5/5! + 18380*x^6/6! + 178964*x^7/7! + 1945196*x^8/8! + 23401268*x^9/9! + 309227468*x^10/10! + 4455370196*x^11/11! + 69515708204*x^12/12! + 1167480739700*x^13/13! + 20995796916620*x^14/14! + 402535779740180*x^15/15! + 8195904238269164*x^16/16! +...
where C(x)^2 - 2*A(x)^2 = exp(2*x).

Links

Crossrefs

Cf. A292181 (A), A292183 (C).

Programs

  • PARI
    {a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(B,n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. B(x) and related functions A(x) and C(x) satisfy:
(1a) A(x)^2 + B(x)^2 = C(x)^2.
(1b) B(x)^2 - A(x)^2 = exp(x)^2.
(1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
(2a) A(x) = Integral A(x) + B(x)*C(x) dx.
(2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
(2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
(3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
(3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
(3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
(3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
(4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
(4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
(4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
Limit A292181(n)/A292182(n) = 1.
Limit A292183(n)/A292182(n) = sqrt(2).

A292183 E.g.f. C(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that C'(x) = C(x) + 2*A(x)*B(x).

Original entry on oeis.org

1, 1, 3, 13, 63, 361, 2499, 20581, 196311, 2116561, 25357563, 333765037, 4787007855, 74323701817, 1242253733619, 22243082373301, 424815246293319, 8620744969300321, 185235767397027627, 4201390722798810493, 100309092062158564959, 2514646421630798317897, 66041388198395188082595, 1813259146315114344920581, 51950114633383773360554679, 1550392693763071812557794801, 48120508780248064233484223067
Offset: 0

Views

Author

Paul D. Hanna, Sep 10 2017

Keywords

Comments

Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.
Another Pythagorean triple is the e.g.f.s of A289695, A193543, and A153302, which are related to the Lemniscate sine and cosine functions, sl(x) and cl(x).

Examples

			E.g.f.: C(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 63*x^4/4! + 361*x^5/5! + 2499*x^6/6! + 20581*x^7/7! + 196311*x^8/8! + 2116561*x^9/9! + 25357563*x^10/10! + 333765037*x^11/11! + 4787007855*x^12/12! + 74323701817*x^13/13! + 1242253733619*x^14/14! + 22243082373301*x^15/15! + 424815246293319*x^16/16! +...
where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
RELATED SERIES.
A(x) = x + 3*x^2/2! + 10*x^3/3! + 45*x^4/4! + 259*x^5/5! + 1806*x^6/6! + 14827*x^7/7! + 140367*x^8/8! + 1504576*x^9/9! + 17972559*x^10/10! + 236275711*x^11/11! + 3387012720*x^12/12! + 52572376669*x^13/13! + 878552787927*x^14/14! + 15729439074058*x^15/15! + 300400031036745*x^16/16! +...
where A(x) = Integral A(x) + B(x)*C(x) dx.
B(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 35*x^4/4! + 226*x^5/5! + 1715*x^6/6! + 14701*x^7/7! + 141248*x^8/8! + 1515661*x^9/9! + 18048527*x^10/10! + 236581984*x^11/11! + 3386091821*x^12/12! + 52533799501*x^13/13! + 877993866290*x^14/14! + 15723411375931*x^15/15! + 300349139257727*x^16/16 +...
where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
Squares of series.
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 134*x^4/4! + 1050*x^5/5! + 9158*x^6/6! + 89418*x^7/7! + 972470*x^8/8! + 11700378*x^9/9! + 154613222*x^10/10! + 2227684074*x^11/11! + 34757852054*x^12/12! + 583740365754*x^13/13! + 10497898450118*x^14/14! + 201267889853706*x^15/15! + 4097952119101814*x^16/16! +...
where A(x)^2 + B(x)^2 = C(x)^2.
B(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 150*x^4/4! + 1082*x^5/5! + 9222*x^6/6! + 89546*x^7/7! + 972726*x^8/8! + 11700890*x^9/9! + 154614246*x^10/10! + 2227686122*x^11/11! + 34757856150*x^12/12! + 583740373946*x^13/13! + 10497898466502*x^14/14! + 201267889886474*x^15/15! + 4097952119167350*x^16/16! +...
where B(x)^2 - A(x)^2 = exp(2*x).
C(x)^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 284*x^4/4! + 2132*x^5/5! + 18380*x^6/6! + 178964*x^7/7! + 1945196*x^8/8! + 23401268*x^9/9! + 309227468*x^10/10! + 4455370196*x^11/11! + 69515708204*x^12/12! + 1167480739700*x^13/13! + 20995796916620*x^14/14! + 402535779740180*x^15/15! + 8195904238269164*x^16/16! +...
where C(x)^2 - 2*A(x)^2 = exp(2*x).

Links

Crossrefs

Cf. A292181 (A), A292182 (B).

Programs

  • PARI
    {a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(C,n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. C(x) and related functions A(x) and B(x) satisfy:
(1a) A(x)^2 + B(x)^2 = C(x)^2.
(1b) B(x)^2 - A(x)^2 = exp(x)^2.
(1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
(2a) A(x) = Integral A(x) + B(x)*C(x) dx.
(2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
(2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
(3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
(3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
(3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
(3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
(4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
(4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
(4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
Limit A292183(n)/A292181(n) = sqrt(2).
Limit A292183(n)/A292182(n) = sqrt(2).
Showing 1-9 of 9 results.

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