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Triangle read by rows: row n is the expansion of x^n in terms of (x+k)!/x! for decreasing k.
1

%I #12 Jun 22 2012 15:50:31

%S 1,1,-1,1,-3,1,1,-6,7,-1,1,-10,25,-15,1,1,-15,65,-90,31,-1,1,-21,140,

%T -350,301,-63,1,1,-28,266,-1050,1701,-966,127,-1,1,-36,462,-2646,6951,

%U -7770,3025,-255,1,1,-45,750,-5880,22827,-42525,34105,-9330,511,-1

%N Triangle read by rows: row n is the expansion of x^n in terms of (x+k)!/x! for decreasing k.

%C Signed version of A008278.

%e Triangle starts

%e 1,

%e 1, -1,

%e 1, -3, 1,

%e 1, -6, 7, -1,

%e 1, -10, 25, -15, 1,

%e 1, -15, 65, -90, 31, -1,

%e ...

%e The fourth row corresponds to the expansion

%e x^4 = 1 * (x + 3)! / x! - 6 * (x + 2)! / x! + 7 * (x + 1)! / x! - 1.

%o (Maxima) f1(p, n) := if n = 0 then [p] else [coeff(p, x ^ n), f1(p - expand(product(x + i, i, 1, n) * coeff(p, x^ n)), n - 1)$

%o f(n) := flatten(f1(x ^ n, n))$

%K sign,tabl

%O 0,5

%A _Douglas Boffey_, Jun 18 2012