A112708 -id:A112708 - cp's OEIS frontend

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

Wolfdieter Lang, Oct 31 2005

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.

W. Lang: First 10 rows.

Row sums give A112708. Unsigned row sums give A112709.

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

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

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.

A112709 Unsigned row sums of triangle A112707 (partial sums of Catalan numbers multiplied by powers of negative numbers).

Original entry on oeis.org

1, 2, 2, 5, 14, 65, 376, 2613, 20761, 184756, 1816330, 19526217, 227588213, 2855033988, 38311846704, 547128228043, 8279365967056, 132259968568859, 2223048616707842, 39199883209336095, 723269314782624612

Offset: 0

Wolfdieter Lang, Oct 31 2005

Cf. A112707, A112708 (signed row sums).

a(n) = Sum_{m=0..n} |A112707(n, m)|, for n >= 0.

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cp's OEIS Frontend

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

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