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

%I A100640 #15 Nov 12 2018 07:20:07
%S A100640 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,
%T A100640 751,3577,49,2989,2989,49,3577,751,989,2944,-464,5248,-454,5248,-464,
%U A100640 2944,989,2857,15741,27,1209,2889,2889,1209,27,15741,2857,16067,26575,-16175,5675
%N A100640 Triangle read by rows: numerators of Cotesian numbers C(n,k) (0 <= k <= n).
%D A100640 Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
%D A100640 L. M. Milne-Thompson, Calculus of Finite Differences, MacMillan, 1951, p. 170.
%p A100640 (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 A100640 # Another program:
%p A100640 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 A100640 [[1], seq( [seq(T(n, k), k=1..n)], n=2..14)];
%t A100640 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]&
%t A100640 (* _Jean-François Alcover_, May 17 2011, after Maple prog. *)
%Y A100640 Cf. A100641-A100648, A100620, A100621, A002177, A002176.
%K A100640 sign,frac,tabl
%O A100640 0,5
%A A100640 _N. J. A. Sloane_, Dec 04 2004