A067078 -id:A067078 - cp's OEIS frontend

A130045 Denominator of polynomial a[1]=1, a[2]->1+1/(xa[1]), a[3]->1+1/(2xa[2]), a[4]->1+1/(3xa[3]),.. giving 1,(1+x)/x,(3+2x)/(2(1+x)),(2+11x+6x^2)/(3x(3+2x)), .. at x-> -1. Absolute values are equal to A067078(n)/n.

1, 1, 4, -9, -20, 55, 210, -1085, -7000, 53235, 462350, -4500265, -48454980, 571411295, 7321388410, -101249656725, -1502852293040, 23827244817355, 401839065437670, -7182224591785985, -135607710526966300, 2696935204638786615, 56349204870460046930, -1234002202313888987245

Offset: 1

Wouter Meeussen, May 02 2007

The iterated form (see Mathematica line) links some seemingly disparate sequences.

Cf. A067078, A052169, A002467, A001053, A093858.

Cf. A003422

Denominator[Together[k=1;NestList[1+1/((k++)x #)&,x,24]]]/.x->(-1)

A067078 has recurrence a(1) = 1, a(2) = 2, a(n) = (n-1)*a(n-1) - (n-2)*a(n-2).

abs(a(n))=(n-1)*sum(k!,k=0..n-3)+(n-1), n>1. [From Gary Detlefs, Feb 05 2011]

A014288 a(n) = floor(Sum_{k=0..n} k!/2), or floor( A003422(n+1)/2 ).

Original entry on oeis.org

0, 1, 2, 5, 17, 77, 437, 2957, 23117, 204557, 2018957, 21977357, 261478157, 3374988557, 46964134157, 700801318157, 11162196262157, 189005910310157, 3390192763174157, 64212742967590157, 1280663747055910157, 26826134832910630157, 588826498721714470157

Offset: 0

N. J. A. Sloane

nonn

The first term a(0) would be a fraction if the floor( ... ) function were omitted; for n >= 2, all terms from A003422 are even. - M. F. Hasler, Dec 16 2007

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A003422, A007489, A067078.

[Floor((&+[Factorial(j): j in [0..n]])/2): n in [0..30]]; // G. C. Greubel, Sep 05 2022

a:= proc(n) a(n):= `if`(n<3, n, (n+1)*a(n-1)-n*a(n-2)) end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 01 2013

f[x_] := {Floor[1 + (n - x[[2]])*x[[1]]], x[[2]] + 1};
a[0] = 0; a[n_] := Nest[f, {1, 0}, n][[1]]/2 (* Joseph E. Cooper III (easonrevant(AT)gmail.com), Aug 19 2008 *) (* updated by Jean-François Alcover, Jun 01 2015 *)
a[n_]:=-(1/2) Subfactorial[-1]-1/2(-1)^n Gamma[2+n] Subfactorial[-2-n]; Table[a[n] //FullSimplify,{n,0,25}] (* Gerry Martens, May 29 2015 *)

A014288(n)=sum(k=0,n,k!)>>1 \\ M. F. Hasler, Dec 16 2007

from math import factorial
def A014288(n): return sum(factorial(k) for k in range(n+1))>>1 # Chai Wah Wu, Nov 01 2023

[sum(factorial(j) for j in (0..n))//2 for n in (0..30)] # G. C. Greubel, Sep 05 2022

a(0)=0, a(1)=1, a(2)=2, a(n) = (n+1)*a(n-1) - n*a(n-2). - Benoit Cloitre, Sep 07 2002
a(0) = 0, a(n) = (1/2)*floor(1 + 1*floor(1 + 2*floor(1 + ... + (n-1)*floor(1+n*floor(1))). - Joseph E. Cooper III (easonrevant(AT)gmail.com), Aug 19 2008
G.f.: G(0)/(1-x)/2 -1/2, where G(k)= 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: A(x) = (Sum_{n>=0} x^n*n!)/(2-2*x) - 1/2 = G(0)/(4*(1-x)) - 1/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
a(n) ~ n!/2. - Vaclav Kotesovec, Aug 10 2013
E.g.f.: -1/2 + (exp(x)/2)*Sum_{k>=0} (k! - k*Gamma(k,x)). - Robert Israel, Jun 01 2015
a(n) = ((n+1)!*ExpIntegral(n+2,-1)+Ei(1)+Pi*i)/(2*e). - Ammar Khatab, Aug 14 2020

Edited by M. F. Hasler, Dec 16 2007

A165680 Triangle of the divisors of the coefficients of triangles A138771 and A165675.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 1, 2, 6, 24, 1, 1, 1, 2, 6, 24, 120, 1, 1, 1, 2, 6, 24, 120, 720, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880

Offset: 1

Johannes W. Meijer, Oct 05 2009

			Triangle starts:
1,
1, 1,
1, 1, 1,
1, 1, 1, 2,
1, 1, 1, 2, 6,
1, 1, 1, 2, 6, 24,
1, 1, 1, 2, 6, 24, 120,
1, 1, 1, 2, 6, 24, 120, 720,
1, 1, 1, 2, 6, 24, 120, 720, 5040,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880,
...

Cf. A138771, A165675 and A000142.
A000012 (3x), A007395, A010722, A010863 equal the first six left hand columns.
A159333 equals, for n=>-1, all right hand columns.
A067078 equals the row sums.

nmax:=11: for n from 1 to nmax do a(n,1):=1 od: for n from 2 to nmax do for m from 2 to n do a(n,m):=(m-2)! od: od: for n from 1 to nmax do seq(a(n,m),m=1..n) od;

a(n) = A138771(n)/A165675(n-1).

cp's OEIS Frontend

A130045 Denominator of polynomial a[1]=1, a[2]->1+1/(xa[1]), a[3]->1+1/(2xa[2]), a[4]->1+1/(3xa[3]),.. giving 1,(1+x)/x,(3+2x)/(2(1+x)),(2+11x+6x^2)/(3x(3+2x)), .. at x-> -1. Absolute values are equal to A067078(n)/n.

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A014288 a(n) = floor(Sum_{k=0..n} k!/2), or floor( A003422(n+1)/2 ).

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A165680 Triangle of the divisors of the coefficients of triangles A138771 and A165675.

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cp's OEIS Frontend

A130045 Denominator of polynomial a[1]=1, a[2]->1+1/(x*a[1]), a[3]->1+1/(2*x*a[2]), a[4]->1+1/(3*x*a[3]),.. giving 1,(1+x)/x,(3+2*x)/(2*(1+x)),(2+11*x+6*x^2)/(3*x*(3+2*x)), .. at x-> -1. Absolute values are equal to A067078(n)/n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A014288 a(n) = floor(Sum_{k=0..n} k!/2), or floor( A003422(n+1)/2 ).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

SageMath

Formula

Extensions

A165680 Triangle of the divisors of the coefficients of triangles A138771 and A165675.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A130045 Denominator of polynomial a[1]=1, a[2]->1+1/(xa[1]), a[3]->1+1/(2xa[2]), a[4]->1+1/(3xa[3]),.. giving 1,(1+x)/x,(3+2x)/(2(1+x)),(2+11x+6x^2)/(3x(3+2x)), .. at x-> -1. Absolute values are equal to A067078(n)/n.