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 A112707 Triangle built from partial sums of Catalan numbers multiplied by powers of nonpositive numbers. 12

%I

%S 1,1,1,1,0,1,1,2,-1,1,1,-3,7,-2,1,1,11,-33,16,-3,1,1,-31,191,-119,29,

%T -4,1,1,101,-1153,1015,-291,46,-5,1,1,-328,7295,-9191,3293,-579,67,-6,

%U 1,1,1102,-47617,87037,-39715,8171,-1013,92,-7,1,1,-3760,318463,-851186,500957,-123079,17131,-1623,121

%N Triangle built from partial sums of Catalan numbers multiplied by powers of nonpositive numbers.

%C The column sequences (without leading zeros) begin with A000012 (powers of 1), A032357(n)*(-1)^n, A064306(n)*(-1)^n, A112710, A112711, A113264-A113269, for m=0.. 10.

%H W. Lang: <a href="http://www.itp.kit.edu/~wl/EISpub/A112707.text">First 10 rows.</a>

%F a(n, m)=sum(C(k)*(-m)^k, k=0..n-m), with C(k):=A000108(k) (Catalan) if n>m>0; a(n, n)=1, a(n, 0)=1, n>=0; a(n, m)=0 if n<m.

%F G.f. for column m>=0 (without leading zeros): c(-m*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.

%Y Row sums give A112708. Unsigned row sums give A112709.

%Y Cf. A112705 (similar triangle with powers of positive numbers).

%K sign,easy,tabl

%O 0,8

%A _Wolfdieter Lang_, Oct 31 2005

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