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.

A141055 The n-th differences of the row A141045(n,.).

Original entry on oeis.org

-1, -2, -60, -120, -5040, -10080, -302400, -604800, -39916800, -79833600, -217945728000, -435891456000, -2615348736000, -5230697472000, -2667655710720000, -5335311421440000, -4257578514309120000, -8515157028618240000, -2810001819444019200000
Offset: 0

Views

Author

Paul Curtz, Aug 01 2008

Keywords

Comments

Can be thought of as the second sequence of a family: the first is A091137, the third starts 1, 2, 84, 168.

Examples

			The first differences of A141045(2,.) are 11-(-19)=30 and -19-11 = -30. The 2nd difference is a(2)= -30-30= -60.

Programs

  • Mathematica
    a[n_] := a[n] = a[n-1]*If[OddQ[n], 2, Denominator[BernoulliB[n+2]]]; a[0]=-1; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 17 2014 *)
  • PARI
    a(n)=if(n,my(pr=a(n-1));fordiv(n+2,d,if(isprime(d+1),pr*=d+1));pr,-1) \\ Charles R Greathouse IV, Jul 08 2011

Formula

a(n+1) = a(n)*A027760(n+3).

Extensions

Edited and extended by R. J. Mathar, Aug 12 2008

A141047 Numerators of A091137(n)*T(n,n)/n! where T(i,j)=Integral (x= i.. i+1) x*(x-1)*(x-2)* .. *(x-j+1) dx.

Original entry on oeis.org

1, 3, 23, 55, 1901, 4277, 198721, 434241, 14097247, 30277247, 2132509567, 4527766399, 13064406523627, 27511554976875, 173233498598849, 362555126427073, 192996103681340479, 401972381695456831, 333374427829017307697, 691668239157222107697, 236387355420350878139797
Offset: 0

Views

Author

Paul Curtz, Jul 31 2008

Keywords

Comments

Numerators of the main diagonal of the array A091137(j)*T(i,j)/j! where T(i,j)=Integral (x= i.. i+1) x*(x-1)*(x-2)* .. *(x-j+1) dx.
The reduced fractions of the array T(i,j) are shown in A140825, which also describes how the integrand is a generating function of Stirling numbers.
The sequence A027760 plays a role i) in relating to A091137 as described there and
ii) in a(n+1)-A027760(n+1)*a(n)= A002657(n+1), numerators of the diagonal T(n,n+1).

References

  • P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note 12, Centre de Calcul Scientifique de l' Armement, Arcueil (1969), p. 36.

Crossrefs

Cf. A140811, A140825, A141045.

Programs

  • Maple
    T := proc(i,j) local var,k ; var := x ; for k from 1 to j-1 do var := var*(x-k) ; od: int(var,x=i..i+1) ; simplify(A091137(j)*%/j!) ; numer(%) ; end:
A141047 := proc(n) T(n,n) ; end: for n from 0 to 20 do printf("%a,",A141047(n) ) ; od: # R. J. Mathar, Feb 23 2009
  • Mathematica
    b[n_] := b[n] = (* A091137 *) If[n==0, 1, Product[d, {d, Select[Divisors[n] + 1, PrimeQ]}]*b[n-1]]; T[i_, j_] := Integrate[Product[x-k, {k, 0, j-1}], {x, i, i+1}]; a[n_] := b[n]*T[n, n]/n!; Table[a[n] // Numerator, {n, 0, 20}] (* Jean-François Alcover, Jan 10 2016 *)

Formula

a(n) = numerator( A091137(n)*T(n,n)/n!) where T(n,n) = sum_{k=0..n} A048994(n,k)*((n+1)^(k+1)-n^(k+1))/(k+1).

Extensions

Edited and extended by R. J. Mathar, Feb 23 2009
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.