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 1-3 of 3 results.

A000828 E.g.f. cos(x)/(cos(x) - sin(x)).

Original entry on oeis.org

1, 1, 2, 8, 40, 256, 1952, 17408, 177280, 2031616, 25866752, 362283008, 5535262720, 91620376576, 1633165156352, 31191159799808, 635421069967360, 13753735117275136, 315212388819402752, 7625476699018231808
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

For a refinement of these numbers see A185896.
A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...|x_n|} = {1,2...,n}. Let x_1,...,x_n be a signed permutation. Then we say 0,x_1,...,x_n,0 is a snake of type S(n;0,0) when 0 < x_1 > x_2 < ... 0. For example, 0 4 -3 -1 -2 0 is a snake of type S(4;0,0). Then a(n) equals the cardinality of S(n;0,0) [Verges]. An example is given below. - Peter Bala, Sep 02 2011
Original name was: E.g.f. cos(x)*(cos(x)+sin(x)) /cos(2*x). - Arkadiusz Wesolowski, Jul 25 2012
Number of plane (that is, ordered) increasing 0-1-2 trees on n vertices where the vertices of outdegree 1 or 2 come in two colors. An example is given below. - Peter Bala, Oct 10 2012

Examples

			a(3) = 8: The eight snakes of type S(3;0,0) are
0 1 -2 3 0, 0 1 -3 2 0, 0 2 1 3 0, 0 2 -1 3 0, 0 2 -3 1 0,
0 3 1 2 0, 0 3 -1 2 0, 0 3 -2 1 0.
1 + x + 2*x^2 + 8*x^3 + 40*x^4 + 256*x^5 + 1952*x^6 + 17408*x^7 + ...
a(3) = 8: The eight increasing 0-1-2 trees on 3 vertices are
..1o (x2 colors)......1o (x2 colors)......1o (x2 colors).....
...|................./.\................./.\.................
..2o (x2 colors)...2o...o3.............3o...o2...............
...|
..3o
Totals.......................................................
...4......+...........2.........+.........2....=...8.........

Links

Crossrefs

Cf. A000464, A000825. A000111, A185896, A235131, A235132.

Programs

  • Maple
    A000828 := n -> (-1)^((n-1)*n/2)*4^n*(Euler(n,1/2)+Euler(n,1))/2: # Peter Luschny, Nov 25 2010
  • Mathematica
    a[n_] := (-1)^((n-1)*n/2)*4^n*(EulerE[n, 1/2] + EulerE[n, 1])/2; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 22 2012, after Peter Luschny *)
  • Maxima
    a(n):=sum(if evenp(n+k) then (-1)^((n+k)/2)*sum(j!/n!*stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n),k,1,n); /* Vladimir Kruchinin, Aug 18 2010 */
  • PARI
    my(x='x + O('x^30)); Vec(serlaplace(cos(x)/(cos(x)-sin(x)))) \\ Michel Marcus, Nov 21 2020

Formula

E.g.f.: 1/(1- tan(x)). - Emeric Deutsch, Sep 10 2001
a(n) = A000831(n)/2 for n>0. - Peter Luschny, Nov 25 2010
a(n) = Sum_{k=1..n, n+k is even} (-1)^((n+k)/2)*Sum_{j=k..n} j!/n!*Stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1,k-1), n>0. - Vladimir Kruchinin, Aug 18 2010
a(n) = (-1)^((n^2-n)/2)*4^n*(E_{n}(1/2)+E_{n}(1))/2 for n >= 0, where E_{n}(x) is an Euler polynomial. - Peter Luschny, Nov 25 2010
From Peter Bala, Sep 02 2011: (Start)
a(n) = (2*i)^(n-1)*Sum_{k = 1..n} (-1)^(n-k)*k!* Stirling2(n,k) * ((1-i)/2)^(k-1), where i = sqrt(-1).
a(n) = 2^(n-1)*A000111(n) for n >= 1.
Let f(x) = 1+x^2 and define the effect of the operator D on a function g(x) by D(g(x)) = d/dx(f(x)*g(x)). Then for n >= 0, a(n+1) = D^n(1) evaluated at x = 1. (End)
From Sergei N. Gladkovskii, Dec 09 2011 - Dec 23 2013: (Start) Continued fractions:
E.g.f.: 1 + x/(G(0)-x); G(k) = 2*k + 1 - (x^2)/G(k+1).
E.g.f.: 1 + x/(U(0)-2*x) where U(k) = 4*k+1 + x/(1+x/(4*k+3 - x/(1- x/U(k+1)))).
E.g.f.: 1 + x/(U(0)-x) where U(k) = 2*k+1 - x^2/U(k+1).
G.f.: 1 + x/G(0) where G(k) = 1 - x*(2*k+2) - 2*x^2*(k+1)*(k+2)/G(k+1).
E.g.f.: 1 + x/T(0) where T(k) = 4*k+1 - x/(1 - x/(4*k+3 + x/(1 + x/T(k+1)))).
G.f.: 1 + x/Q(0) where Q(k) = 1 - 2*x*(2*k+1) - 2*x^2*(2*k+1)*(2*k+2)/(1 - 2*x*(2*k+2) - 2*x^2*(2*k+2)*(2*k+3)/Q(k+1)).
G.f.: 1 + x/(1-2*x)*T(0) where T(k) = 1 - 2*x^2*(k+1)*(k+2)/( 2*x^2*(k+1)*(k+2) - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1)).
E.g.f.: T(0) where T(k) = 1 + x/(4*k+1 - x/(1 - x/( 4*k+3 + x/T(k+1)))). (End)
G.f.: 1 /(1 - 1*x /(1 - 1*x /(1 - 4*x /(1 - 2*6*x^2 /(1 - 6*x /(1 - 4*x /(1 - 4*x /(1 - 10*x /(1 - 5*12*x^2 /(1 - 12*x / ...)))))))))). - Michael Somos, May 12 2012
a(n) ~ n! * 2^(2*n+1)/Pi^(n+1). - Vaclav Kotesovec, Jun 21 2013
a(0) = a(1) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k-1). - Ilya Gutkovskiy, Nov 21 2020
From Peter Bala, Dec 04 2021: (Start)
F(x) = exp(2*x)*(exp(2*x) - 1)/(exp(4*x) + 1) = x + 2*x^2/2! - 8*x^3/3! - 40*x^4/4! + 256*x^5/5! + 1952*x^6/6! - - + + ... is the e.g.f. for the sequence [1, 2, -8, -40, 256, 1952, ...], a signed version of this sequence without the first term.
Let G(x) = x + 2*x^2 - 8*x^3 - 40*x^4 + 256*x^5 + 1952*x^6 - - + + ... be the corresponding o.g.f. We have the continued fraction representation G(x) = x/(1 - 2*x + 12*x^2/(1 + 20*x^2/(1 - 2*x + 56*x^2/(1 + 72*x^2/(1 - 2*x + ... + 4*n*(4*n-1)*x^2/(1 + 4*n*(4*n+1)*x^2/(1 - 2*x + ... ))))))).
The inverse binomial transform 1/(1 + x)*G(x/(1 + x)) = x - 11*x^3 + 361*x^5 - 24611*x^7 + - ... is a g.f. for a signed and aerated version of A000464. (End)

Extensions

Name changed by Arkadiusz Wesolowski, Jul 25 2012

A235134 Expansion of e.g.f. 1/(1 - sinh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 19, 153, 1561, 19563, 289339, 4932273, 95258161, 2055639123, 49019157859, 1280056939593, 36329281202761, 1113449691889083, 36651273215389579, 1289577677407798113, 48299079453732363361, 1918528841276621473443, 80559757274836073592499
Offset: 0

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Author

Vaclav Kotesovec, Jan 03 2014

Keywords

Comments

Generally, for e.g.f. 1/(1-sinh(p*x))^(1/p) we have a(n) ~ n! * p^n / (Gamma(1/p) * 2^(1/(2*p)) * n^(1-1/p) * (arcsinh(1))^(n+1/p)).

Links

Crossrefs

Cf. A006154, A235135.
Cf. A001147, A001586, A136630, A235131, A380155.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-Sinh[2*x])^(1/2), {x, 0, 20}], x] * Range[0, 20]!
  • PARI
    x='x+O('x^50); Vec(serlaplace(1/(sqrt(1-sinh(2*x))))) \\ G. C. Greubel, Apr 05 2017
  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*2^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

Formula

a(n) ~ n! * 2^(n-1/4) / (sqrt(Pi*n) * (log(1+sqrt(2)))^(n+1/2)).
a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A136630(n,k). - Seiichi Manyama, Jun 24 2025

A235132 E.g.f. 1/(1 - tan(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 46, 568, 9976, 203104, 4995136, 140343808, 4493656576, 160609429504, 6356981099776, 275688520680448, 13008983675954176, 663382602064482304, 36360098005522825216, 2131554196360938815488, 133093201551208236875776, 8818123347826691244949504
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 03 2014

Keywords

Comments

Generally, for e.g.f. 1/(1-tan(p*x))^(1/p) is a(n) ~ n! * 2^(2*n+1/p) * n^((1-p)/p) * p^n / (Pi^(n+1/p) * Gamma(1/p)).

Crossrefs

Cf. A000828 (p=1), A235131 (p=2).

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - Tan[3*x])^(1/3), {x, 0, 20}], x] * Range[0, 20]!

Formula

a(n) ~ n! * 2^(2*n+1/3) * 3^n / (Gamma(1/3) * Pi^(n+1/3) * n^(2/3)).
Showing 1-3 of 3 results.

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

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