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.

A002793 a(n) = 2n*a(n-1) - (n-1)^2*a(n-2).

Original entry on oeis.org

0, 1, 4, 20, 124, 920, 7940, 78040, 859580, 10477880, 139931620, 2030707640, 31805257340, 534514790680, 9591325648580, 182974870484120, 3697147584561340, 78861451031150840, 1770536585183202980, 41729280102868841080, 1030007496863617367420, 26568602827124392999640
Offset: 0

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Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

From Wolfdieter Lang, Dec 12 2011: (Start)
r(n) = a(n+1)*(-1)^n, n >= 0, gives the alternating row sums of the coefficient triangle A199577, i.e., r(n)=La_n(1;0,-1), with the monic first associated Laguerre polynomials with parameter alpha=0 evaluated at x=-1.
The e.g.f. for these row sums r(n) is g(x) = -(2+x)*exp(1/(1+x))*(Ei(1,1/(1+x))-Ei(1,1))/(1+x)^3 + 1/(1+x)^2, with the exponential integral Ei(1,x) = Gamma(0,x).
This e.g.f. satisfies the homogeneous ordinary second-order differential equation (1+x)^2*(d^2/dx^2)g(x) + (6+5*x)*(d/dx)g(x) + 4*g(x) = 0, g(0)=1, (d/dx)g(x)|_{x=0}=-4.
This e.g.f. g(x) is equivalent to the recurrence
b(n)= -2*(n+1)*b(n-1) - n^2*b(n-2), b(-1)=0, b(0)=1.
Therefore, the e.g.f. of a(n) is A(x)=int(g(-x),x), with A(0)=0. This agrees with the e.g.f. given below in the formula section by Max Alekseyev.
(End)

References

  • J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 356.

Links

Crossrefs

Bisection of A056952. A199577 (alternating row sums, unsigned).
Cf. A002720, A073003.

Programs

  • Magma
    I:=[1, 4]; [0] cat [n le 2 select I[n] else 2*n*Self(n-1) - (n-1)^2*Self(n-2): n in [1..30]]; // G. C. Greubel, May 16 2018
  • Mathematica
    Flatten[{0,RecurrenceTable[{(-1+n)^2 a[-2+n]-2 n a[-1+n]+a[n]==0,a[1]==1,a[2]==4}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 19 2013 *)
nxt[{n_,a_,b_}]:={n+1,b,2(n+1)b-n^2 a}; NestList[nxt,{1,0,1},30][[All,2]] (* Harvey P. Dale, Sep 06 2022 *)
  • PARI
    A058006(n) = sum(k=0,n, (-1)^k*k! );
a(n) = if (n<=1, n, sum(k=1, n, (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)! ) ); /* Joerg Arndt, Oct 12 2012 */
  • PARI
    {a(n)=if(n==1,1,polcoeff(1-sum(m=1, n-1, a(m)*x^m*(1-(m+1)*x+x*O(x^n))^2), n))} \\ Paul D. Hanna, Feb 06 2013

Formula

From Max Alekseyev, Jul 06 2010: (Start)
For n > 1, a(n) = Sum_{k=1..n} (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)!.
E.g.f.: (Gamma(0,1) - Gamma(0,1/(1-x))) * exp(1/(1-x)) / (1-x). (End)
From Peter Bala, Oct 11 2012: (Start)
Numerators in the sequence of convergents of Stieltjes's continued fraction for A073003, the Euler-Gompertz constant G := int {x = 0..oo} 1/(1+x)*exp(-x) dx:
G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The denominators are in A002720.
(End)
G.f.: x = Sum_{n>=1} a(n) * x^n * (1 - (n+1)*x)^2. - Paul D. Hanna, Feb 06 2013
a(n) ~ G * exp(2*sqrt(n) - n - 1/2) * n^(n+1/4) / sqrt(2) * (1 + 31/(48*sqrt(n))), where G = 0.596347362323194... is the Gompertz constant (see A073003). - Vaclav Kotesovec, Oct 19 2013

Extensions

Edited by Max Alekseyev, Jul 13 2010

A056922 Denominators of continued fraction for alternating factorial.

Original entry on oeis.org

1, 1, 2, 3, 8, 11, 41, 52, 249, 301, 1754, 2055, 14084, 16139, 127057, 143196, 1272625, 1415821, 14015014, 15430835, 168323364, 183754199, 2189619553, 2373373752, 30670104577, 33043478329, 460235322854, 493278801183
Offset: 0

Views

Author

Aleksandar Petojevic, Sep 05 2000

Keywords

Links

Crossrefs

Cf. A058799, A056952 (numerators).

Formula

a(0)=1; a(1)=1; a(2*n) = n*a(2*n-1) + a(2*n-2); a(2*n+1) = a(2*n) + a(2*n-1).

Extensions

More terms from James Sellers, Sep 07 2000

A249138 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 6, 5, 7, 1, 1, 6, 18, 8, 10, 1, 1, 24, 26, 46, 12, 14, 1, 1, 24, 96, 58, 86, 16, 18, 1, 1, 120, 154, 326, 118, 156, 21, 23, 1, 1, 120, 600, 444, 756, 198, 246, 26, 28, 1, 1, 720, 1044, 2556, 1152, 1692, 324, 384, 32, 34, 1, 1
Offset: 0

Views

Author

Clark Kimberling, Oct 22 2014

Keywords

Comments

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + floor((n+2)/2)/f(n-1,x), where f(0,x) = 1.
(Sum of numbers in row n) = A056952(n+2) for n >= 0.
(Column 1) is essentially A081123 (repeated factorials).

Examples

			f(0,x) = 1/1, so that p(0,x) = 1;
f(1,x) = (1 + x)/1, so that p(1,x) = 1 + x;
f(2,x) = (2 + x + x^2)/(1 + x), so that p(2,x) = 2 + x + x^2.
First 6 rows of the triangle of coefficients:
  1
  1    1
  2    1    1
  2    4    1    1
  6    5    7    1    1
  6    18   8    10   1   1

Links

Crossrefs

Cf. A056952, A081123, A249128.

Programs

  • Mathematica
    z = 15; p[x_, n_] := x + Floor[(n+1)/2]/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249138 array *)
Flatten[CoefficientList[u, x]] (* A249138 sequence *)
Showing 1-3 of 3 results.

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