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

1, 2, 2, 6, 3, 6, 8, 8, 8, 8, 90, 45, 15, 45, 90, 288, 96, 144, 144, 96, 288, 840, 35, 280, 105, 280, 35, 840, 17280, 17280, 640, 17280, 17280, 640, 17280, 17280, 28350, 14175, 14175, 14175, 2835, 14175, 14175, 14175, 28350, 89600, 89600, 2240, 5600, 44800, 44800, 5600

Offset: 0

Author

N. J. A. Sloane, Dec 04 2004

Keywords

• nonn
• frac
• tabl

Examples

			Triangle A100640/A100641 begins:
[1],
[1/2, 1/2],
[1/6, 2/3, 1/6],
[1/8, 3/8, 3/8, 1/8],
[7/90, 16/45, 2/15, 16/45, 7/90],
[19/288, 25/96, 25/144, 25/144, 25/96, 19/288],
[41/840, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840],
[751/17280, 3577/17280, 49/640, 2989/17280, 2989/17280, 49/640, 3577/17280, 751/17280],
...
		

References

• Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
• L. M. Milne-Thompson, Calculus of Finite Differences, MacMillan, 1951, p. 170.

Crossrefs

Cf. A100640-A100648, A100620, A100621, A002177, A002176.

Programs

Maple

(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;
# Another program:
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;
[[1], seq( [seq(T(n,k),k=1..n)], n=2..14)];

Mathematica

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] &
(* Jean-François Alcover, May 17 2011, after Maple prog. *)