A100640 Triangle read by rows: numerators of Cotesian numbers C(n,k) (0 <= k <= n).

0, 1, 1, 1, 2, 1, 1, 3, 3, 1, 7, 16, 2, 16, 7, 19, 25, 25, 25, 25, 19, 41, 9, 9, 34, 9, 9, 41, 751, 3577, 49, 2989, 2989, 49, 3577, 751, 989, 2944, -464, 5248, -454, 5248, -464, 2944, 989, 2857, 15741, 27, 1209, 2889, 2889, 1209, 27, 15741, 2857, 16067, 26575, -16175, 5675

Offset

0,5

References

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.

L. M. Milne-Thompson, Calculus of Finite Differences, MacMillan, 1951, p. 170.

Links

Table of n, a(n) for n=0..58.

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 // Numerator //  Take[#, 59]&
(* Jean-François Alcover, May 17 2011, after Maple prog. *)

Crossrefs

Cf. A100641-A100648, A100620, A100621, A002177, A002176.

Sequence in context: A026009 A137171 A010356 * A242779 A215065 A175424

Adjacent sequences: A100637 A100638 A100639 * A100641 A100642 A100643

Keyword

sign,frac,tabl

Author

N. J. A. Sloane, Dec 04 2004

Status

approved