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

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, -4, 1, 1, 101, -1153, 1015, -291, 46, -5, 1, 1, -328, 7295, -9191, 3293, -579, 67, -6, 1, 1, 1102, -47617, 87037, -39715, 8171, -1013, 92, -7, 1, 1, -3760, 318463, -851186, 500957, -123079, 17131, -1623, 121

Offset

0,8

Comments

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.

Links

Table of n, a(n) for n=0..63.

W. Lang: First 10 rows.

Formula

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.

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.

Crossrefs

Row sums give A112708. Unsigned row sums give A112709.

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

Sequence in context: A296526 A362377 A259844 * A196017 A343555 A251660

Adjacent sequences: A112704 A112705 A112706 * A112708 A112709 A112710

Keyword

sign,easy,tabl

Author

Wolfdieter Lang, Oct 31 2005

Status

approved