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 1-4 of 4 results.

A143601 Number of labeled odd-degree trees with 2n+1 nodes.

Original entry on oeis.org

1, 1, 13, 541, 47545, 7231801, 1695106117, 567547087381, 257320926233329, 151856004814953841, 113144789723082206461, 103890621918675777804301, 115270544419577901796226473, 152049571406030636219959644841
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2008, May 27 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! + ...
The e.g.f. of A007106 (a bisection of A058014) is given by:
sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! + ...
The e.g.f. of A058014 is given by:
F(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! + ...
where A(x) = [F(x) + F(-x)]/2 and exp(x*A(x)) = F(x).
The e.g.f. of A143600 is given by:
G(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! + ...
where A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2.

Links

Crossrefs

Cf. A058014, A143600, A007106, A309204.
Cf. A370430, A370432.

Programs

  • Mathematica
    Table[(2*n)!*CoefficientList[1/x*InverseSeries[Series[x/Cosh[x],{x,0,41}],x],x][[2*n+1]],{n,0,20}] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=cosh(x*A));n!*polcoeff(A,n)}
  • PARI
    {a(n)=(2*n)!*polcoeff(cosh(x+x*O(x^(2*n)))^(2*n+1)/(2*n+1),2*n)} \\ Paul D. Hanna, Aug 29 2008
  • PARI
    {a(n) = sum(k=0,n, binomial(2*n+1,k) * (2*n+1-2*k)^(2*n) / ((2*n+1) * 2^(2*n)) )}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Feb 19 2024

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas.
(1) A(x) = cosh(x*A(x)).
(2) A(x) = (1/x)*Series_Reversion( x/cosh(x) ).
(3) sqrt(A(x)^2 - 1) = e.g.f. of A007106.
(4) exp(x*A(x)) = A(x) + sqrt(A(x)^2-1) = e.g.f. of A058014.
(5) A(x) = [F(x) + F(-x)]/2 where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
(6) A(2*x) = [G(x)/G(-x) + G(-x)/G(x)]/2 where G(x) = exp(x*G(x)/G(-x)) = e.g.f. of A143600.
From Paul D. Hanna, Aug 29 2008: (Start)
(7) A(x/cosh(x)) = cosh(x).
(8) a(n) = (2n)!*[x^(2n)] cosh(x)^(2n+1)/(2n+1). (End)
(9) a(n) = Sum_{k=0..n} binomial(2*n+1,k) * (2*n+1 - 2*k)^(2*n) / ((2*n+1) * 2^(2*n)). [See formula by Christophe Vignat in A309204.] - Paul D. Hanna, Feb 19 2024
a(n) ~ 2^(2*n) * n^(2*n-1) * (s^2-1)^(n+1/2) / exp(2*n), where s = 1.810170580698977274512829... is the root of the equation sqrt(s^2-1) * log(s + sqrt(s^2-1)) = s. - Vaclav Kotesovec, Jan 10 2014
Radius of convergence r = 0.66274341934918158097474... = 1/sqrt(s^2-1) and A(r) = s (given above) satisfy r = 1/sinh(r*A(r)) and A(r) = cosh(r*A(r)). - Paul D. Hanna, Mar 04 2024

Extensions

Edited by Paul D. Hanna, May 27 2009

A143599 E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ).

Original entry on oeis.org

1, 1, 3, 10, 53, 316, 2527, 22072, 239689, 2774800, 38284091, 553477024, 9284250109, 161180444608, 3187413648343, 64638167906176, 1473221217774353, 34190645940363520, 882759869810501491, 23079229227696318976
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2008

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...
F(x) = sqrt(A(x)/A(-x)) = e.g.f. of A058014:
F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
where F(x) = exp(x*(F(x) + 1/F(x))/2).
G(x) = [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601:
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
where G(x) = cosh(x*G(x)).
S(x) = [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106:
S(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...
where S(x) = sqrt(G(x)^2 - 1) and G(x) = e.g.f. of A143601.

Links

Crossrefs

Cf. A058014, A143600, A143601, A007106.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x*sqrt(A/subst(A,x,-x))));n!*polcoeff(A,n)}

Formula

E.g.f.: A(x) = exp(x*exp(x*G(x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
E.g.f.: sqrt(A(x)/A(-x)) = F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
E.g.f.: [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601.
E.g.f.: [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106.
E.g.f.: A(x) = H(x/2)^2 where H(x) = exp(x*H(x)/H(-x)) = e.g.f. of A143600.
E.g.f. satisfies: A(x/cosh(x)) = exp(x*exp(x)/cosh(x)). [From Paul D. Hanna, Aug 29 2008]
E.g.f. satisfies: -x*sqrt(x^2/log(y)^2) = log((x^2*y)/log(y)^2), where y=A(x). - Vaclav Kotesovec, Feb 26 2014
a(n) ~ c * n! * d^n / n^(3/2), where d = 1.5088795615383199289... is the root of the equation sqrt(1+1/d^2) = 1 + LambertW((1+sqrt(1+1/d^2))/exp(1 + sqrt(1+1/d^2))), and c = 7.98255033020099890281693169... if n is even, and c = 7.852067808737280621088934789... if n is odd. - Vaclav Kotesovec, Feb 26 2014

A198887 E.g.f. satisfies: A(x) = exp(x*A(x)^2*A(-x)).

Original entry on oeis.org

1, 1, 3, 28, 269, 5056, 84247, 2400448, 57253849, 2191568896, 68151324491, 3278448139264, 125802549088933, 7291045162516480, 332950230966532831, 22581201334925049856, 1196122595530554458033, 92934371464549349982208, 5602230959364892208231443
Offset: 0

Views

Author

Paul D. Hanna, Oct 30 2011

Keywords

Comments

Limit n->infinity (a(n)/n!)^(1/n) = 1/r = 4.263493124332896881178517333221505445574016761952741537891924..., where r = 0.234549457648408586261093274213550311973... and s = 1.724680091765540585933497362883851976875... are roots of the system of equations s*sqrt((r*s*LambertW(2*r*s))/2) = log(s), s*sqrt((r*s*LambertW(2*r*s))/2)*(4 + 3*LambertW(2*r*s)) = 2*(1 + LambertW(2*r*s)). - Vaclav Kotesovec, Jul 16 2014

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 28*x^3/3! + 269*x^4/4! + 5056*x^5/5! +...
Related series:
A(x)^2*A(-x) = 1 + x + 7*x^2/2! + 40*x^3/3! + 709*x^4/4! + 8016*x^5/5! +...
log(A(x)) = x + 2*x^2/2! + 21*x^3/3! + 160*x^4/4! + 3545*x^5/5! + 48096*x^6/6! +...

Links

Crossrefs

Cf. A143600.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(n=0,n,A=exp(x*A^2*subst(A,x,-x)+x*O(x^n)));n!*polcoeff(A,n)}

Formula

E.g.f. satisfies x*y^2*sqrt(LambertW(2*x*y)/(2*x*y)) = log(y), where y = A(x). - Vaclav Kotesovec, Jul 15 2014

A143598 E.g.f.: A(x) = exp(x*sinh(x*G(x))) where G(x) = cosh(x*G(x)) is the e.g.f. of A143601.

Original entry on oeis.org

1, 2, 28, 1176, 103440, 15726880, 3684098496, 1232799974784, 558670427013376, 329559835063067136, 245462725323910487040, 225319148634038399801344, 249936012383478860884217856, 329609037187846742271984869376
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2008

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x^2/2! + 28*x^4/4! + 1176*x^6/6! + 103440*x^8/8! +...
A(x) = exp(x*F(x)) where F(x) = e.g.f. of A007106:
F(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...
A(x) = exp(x*sqrt(G(x)^2 - 1)) where G(x) = e.g.f. of A143601:
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
A(x) = sqrt(H(x)*H(-x)) where H(x) = e.g.f. of A143599:
H(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...

Crossrefs

Cf. A058014, A143600, A143601, A007106.

Programs

  • PARI
    {a(n)=local(G=1+x*O(x^n));for(i=0,n,G=cosh(x*G));n!*polcoeff(exp(x*sqrt(G^2-1)),n)}

Formula

E.g.f.: A(x) = exp(x*F(x)) where F(x) is the e.g.f. of A007106.
E.g.f.: A(x) = sqrt(H(x)*H(-x)) where H(x) = exp(x*sqrt(H(x)/H(-x))) is the e.g.f. of A143599.
E.g.f. satisfies: A(x/cosh(x)) = exp(x*tanh(x)). [From Paul D. Hanna, Aug 29 2008]
Showing 1-4 of 4 results.

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