|
%I #12 Jan 10 2016 10:07:58
%S 1,3,23,55,1901,4277,198721,434241,14097247,30277247,2132509567,
%T 4527766399,13064406523627,27511554976875,173233498598849,
%U 362555126427073,192996103681340479,401972381695456831,333374427829017307697,691668239157222107697,236387355420350878139797
%N 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.
%C 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.
%C 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.
%C The sequence A027760 plays a role i) in relating to A091137 as described there and
%C ii) in a(n+1)-A027760(n+1)*a(n)= A002657(n+1), numerators of the diagonal T(n,n+1).
%D P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note 12, Centre de Calcul Scientifique de l' Armement, Arcueil (1969), p. 36.
%F 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).
%p 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:
%p 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
%t 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 *)
%Y Cf. A140811, A140825, A141045.
%K nonn
%O 0,2
%A _Paul Curtz_, Jul 31 2008
%E Edited and extended by _R. J. Mathar_, Feb 23 2009
|
