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

0,5

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]

Example

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;

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 *)

Crossrefs

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

Sequence in context: A353647 A261790 A174834 * A320438 A255511 A014518

Adjacent sequences: A100639 A100640 A100641 * A100643 A100644 A100645

Keyword

sign,frac,tabl

Author

N. J. A. Sloane, Dec 04 2004

Status

approved