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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). 2
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Apr 11 2013, after Maple *)

CROSSREFS

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

Sequence in context: A010323 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

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Last modified December 12 12:50 EST 2019. Contains 329958 sequences. (Running on oeis4.)