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A100642
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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).
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2
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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
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history;
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OFFSET
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0,5
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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;
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MAPLE
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# (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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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