A112709 -id:A112709 - 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.

A112708 Row sums of triangle A112707 (partial sums of Catalan numbers multiplied by powers of negative numbers).

Original entry on oeis.org

1, 2, 2, 3, 4, -7, 68, -285, 553, 8052, -142982, 1505079, -11345907, 38452956, 672559916, -18995038455, 301298528898, -3473588012773, 23630888402370, 167786255489085, -10244305495490866, 242112502660674867, -4105880426369854764, 48839647963108358039

Offset: 0

Wolfdieter Lang, Oct 31 2005

Cf. A112707, A112709 (unsigned row sums).

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