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%I #26 Nov 12 2018 07:17:32
%S 0,1,1,1,4,1,1,3,3,1,7,32,12,32,7,19,75,50,50,75,19,41,216,27,272,27,
%T 216,41,751,3577,1323,2989,2989,1323,3577,751,989,5888,-928,10496,
%U -4540,10496,-928,5888,989,2857,15741,1080,19344,5778,5778,19344,1080,15741,2857,16067
%N Triangle read by rows: numerators of Cotesian numbers C(n,k) (0 <= k <= n) if the denominators are set to the lcm's of the rows (A002176).
%D Carl Erik Froeberg, Numerical Mathematics, Benjamin/Cummings Pu.Co. 1985, ISBN 0-8053-2530-1, Chapter 17.2.
%D Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
%H Alois P. Heinz, <a href="/A100642/b100642.txt">Rows n = 0..100, flattened</a>
%H W. M. Johnson, <a href="/A002176/a002176.pdf">On Cotesian numbers: their history, computation and values to n=20</a>, Quart. J. Pure Appl. Math., 46 (1914), 52-65. [Annotated scanned copy]
%e 0, 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, ... = A100640/A100641 = A100642/A002176 (the latter is not in lowest terms)
%e Triangle begins
%e 0;
%e 1, 1;
%e 1, 4, 1;
%e 1, 3, 3, 1;
%e 7, 32, 12, 32, 7;
%p # (This defines the Cotesian numbers C(n,i))
%p 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 den:=proc(n) local t1,i; t1:=1; for i from 0 to n do t1:=ilcm(t1,denom(C(n,i))); od: t1; end;
%p # Then den(n)*C(n,k) gives the current sequence
%p seq(seq(den(n,k)*C(n,k), k=0..n), n=0..10);
%t c[n_, i_] /; i == 0 || i == n = (1/n!)*Sum[n^a*StirlingS1[n, a]/(a+1), {a, 1, n+1}]; c[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}]; den[n_] := (For[t1 = 1; i = 0, i <= n, i++, t1 = LCM[t1, c[n, i] // Denominator]]; t1); Table[den[n]*c[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 11 2013, after Maple *)
%Y Cf. A100641, A100620, A100621, A002177, A002176 (row sums), A100640.
%K sign,frac,tabl
%O 0,5
%A _N. J. A. Sloane_, Dec 04 2004
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