A100642 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).

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, 216, 41, 751, 3577, 1323, 2989, 2989, 1323, 3577, 751, 989, 5888, -928, 10496, -4540, 10496, -928, 5888, 989, 2857, 15741, 1080, 19344, 5778, 5778, 19344, 1080, 15741, 2857, 16067

Offset: 0

Author

N. J. A. Sloane, Dec 04 2004

Keywords

• sign
• frac
• tabl

Examples

			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)
Triangle begins
0;
1, 1;
1, 4, 1;
1, 3, 3, 1;
7, 32, 12, 32, 7;
		

References

• Carl Erik Froeberg, Numerical Mathematics, Benjamin/Cummings Pu.Co. 1985, ISBN 0-8053-2530-1, Chapter 17.2.
• Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.

Links

• Alois P. Heinz, Rows n = 0..100, flattened
• W. M. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65. [Annotated scanned copy]

Crossrefs

Cf. A100641, A100620, A100621, A002177, A002176 (row sums), A100640.

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;
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;
# Then den(n)*C(n,k) gives the current sequence
seq(seq(den(n,k)*C(n,k), k=0..n), n=0..10);

Mathematica

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 *)