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 1-2 of 2 results.

A322623 E.g.f.: (1 + sinh(x)) / (1 - sinh(x)).

Original entry on oeis.org

1, 2, 4, 14, 64, 362, 2464, 19574, 177664, 1814162, 20583424, 256891934, 3497611264, 51588733562, 819450793984, 13946142745094, 253171058212864, 4883182404118562, 99727612182790144, 2149854113300939054, 48784173816258494464, 1162353473295706049162, 29013549746780744187904, 757126891483681641073814, 20616734677807356197208064, 584789894473832421848925362
Offset: 0

Views

Author

Paul D. Hanna, Dec 29 2018

Keywords

Comments

Equals the antidiagonal sums of square table A322620.
a(n) = 2*A006154(n) for n >= 1.

Examples

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 64*x^4/4! + 362*x^5/5! + 2464*x^6/6! + 19574*x^7/7! + 177664*x^8/8! + 1814162*x^9/9! + ...
where
A(x) = 1 + 2*sinh(x) + 2*sinh(x)^2 + 2*sinh(x)^3 + 2*sinh(x)^4 + ...

Links

Crossrefs

Cf. A322620, A012261, A006154.

Programs

  • Maple
    S:= series((1+sinh(x))/(1-sinh(x)),x,51):
seq(coeff(S,x,j)*j!,j=0..50);  #  Robert Israel, Dec 31 2018
  • PARI
    {a(n) = my(X = x +x*O(x^n)); n! * polcoeff( (1 + sinh(X)) / (1 - sinh(X)),n)}
for(n=0,30, print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} A322620(n-k,k), for n >= 0.
a(n) ~ sqrt(2)*n!/log(1+sqrt(2))^(n+1). - Robert Israel, Dec 31 2018

A012109 sec(arcsin(sinh(x)))=1+1/2!*x^2+13/4!*x^4+421/6!*x^6+26713/8!*x^8...

Original entry on oeis.org

1, 1, 13, 421, 26713, 2794441, 436186213, 95033434861, 27555582190513, 10260037095841681, 4771143086720391613, 2710025439753915534901, 1846296024220715321941513, 1486014763274444231870834521
Offset: 0

Views

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

Examples

			G.f. = 1 + x + 13*x^2 + 421*x^3 + 26713*x^4 + 2794441*x^5 + ...

Crossrefs

Cf. A012261.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = 2 n + 1}, m! SeriesCoefficient[ EllipticF[ I x, -1] / I, {x, 0, m}]]]; (* Michael Somos, May 05 2017 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ 1 / Sqrt[1 - Sinh[x]^2], {x, 0, m}]]]; (* Michael Somos, May 05 2017 *)
  • PARI
    {a(n) = my(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / sqrt(1 - sinh(x + x * O(x^m))^2), m))}; /* Michael Somos, May 05 2017 */

Formula

From Michael Somos, May 05 2017: (Start)
E.g.f.: Sum_{n>=0} a(n) * x^(2*n) / (2*n)! = sec(arcsin(sinh(x))) = 1 / sqrt(1 - sinh(x)^2).
E.g.f.: Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)! = F(i x| -1) / i where F(phi|m) is the elliptic integral of the 1st kind.
E.g.f. 1 / sqrt(1 - sinh(x)^2) = y satisfies 0 = y''*y + 2*y^2 - 3*y^4 - 3*y'^2 = y - 6*y^3 + 6*y^5 - y''.
a(n) = A012261(2*n). (End)
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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