cp's OEIS Frontend

A100641 Triangle read by rows: denominators of Cotesian numbers C(n,k) (0 <= k <= k).

%I A100641 #15 Apr 24 2014 13:48:59
%S A100641 1,2,2,6,3,6,8,8,8,8,90,45,15,45,90,288,96,144,144,96,288,840,35,280,
%T A100641 105,280,35,840,17280,17280,640,17280,17280,640,17280,17280,28350,
%U A100641 14175,14175,14175,2835,14175,14175,14175,28350,89600,89600,2240,5600,44800,44800,5600
%N A100641 Triangle read by rows: denominators of Cotesian numbers C(n,k) (0 <= k <= k).
%D A100641 Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
%D A100641 L. M. Milne-Thompson, Calculus of Finite Differences, MacMillan, 1951, p. 170.
%e A100641 Triangle A100640/A100641 begins:
%e A100641 [1],
%e A100641 [1/2, 1/2],
%e A100641 [1/6, 2/3, 1/6],
%e A100641 [1/8, 3/8, 3/8, 1/8],
%e A100641 [7/90, 16/45, 2/15, 16/45, 7/90],
%e A100641 [19/288, 25/96, 25/144, 25/144, 25/96, 19/288],
%e A100641 [41/840, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840],
%e A100641 [751/17280, 3577/17280, 49/640, 2989/17280, 2989/17280, 49/640, 3577/17280, 751/17280],
%e A100641 ...
%p A100641 (This defines the Cotesian numbers C(n,i)) with(combinat); C:=proc(n,i) if i=0 or i=n then RETURN( (1/n!)*add(n^a*stirling1(n,a)/(a+1),a=1..n+1) ); fi; (1/n!)*binomial(n,i)* add( add( n^(a+b)*stirling1(i,a)*stirling1(n-i,b)/((b+1)*binomial(a+b+1,b+1)), b=1..n-i+1), a=1..i+1); end;
%p A100641 # Another program:
%p A100641 T:=proc(n,k) (-1)^(n-k)*(n/(n-1))*binomial(n-1,k-1)* integrate(expand(binomial(t-1,n))/(t-k), t=1..n); end;
%p A100641 [[1], seq( [seq(T(n,k),k=1..n)], n=2..14)];
%t A100641 a[n_, i_] /; i == 0 || i == n = 1/n!*Sum[n^a StirlingS1[n, a]/(a + 1), {a, 1, n + 1}]; a[n_, i_] = 1/n!*Binomial[n, i] Sum[n^(a + b)*StirlingS1[i, a]*StirlingS1[n - i, b]/((b + 1)*Binomial[a + b + 1, b + 1]), {b, 1, n}, {a, 1, i + 1}]; Table[a[n, i], {n, 0, 10}, {i, 0, n}] // Flatten // Denominator // Take[#, 52] &
%t A100641 (* _Jean-François Alcover_, May 17 2011, after Maple prog. *)
%Y A100641 Cf. A100640-A100648, A100620, A100621, A002177, A002176.
%K A100641 nonn,frac,tabl
%O A100641 0,2
%A A100641 _N. J. A. Sloane_, Dec 04 2004