cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

Wolfdieter Lang, Oct 31 2005

Keywords

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

Crossrefs

Row sums give A112708. Unsigned row sums give A112709.
Cf. A112705 (similar triangle with powers of positive numbers).

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
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.

A113265 Partial sums of Catalan numbers A000108 multiplied by powers of -6.

Original entry on oeis.org

1, -5, 67, -1013, 17131, -309461, 5849131, -114243413, 2287607467, -46710150485, 968879741611, -20358507992405, 432438339283627, -9270351245202773, 200309903779703467, -4358060643012007253, 95389812498547780267, -2099063396615767545173
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Comments

Seventh column (m=6) of triangle A112707 (partial sums of Catalan numbers multiplied by powers of nonpositive numbers).

Crossrefs

Cf. A113264 (m=5).

Formula

a(n)= A112707(n+6, 6), n>=0.
G.f.: c(-6*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
G.f. for unsigned sequence: c(6*x)/(1+x), hence |a(n)| is the convolution of the sequences {C(n)*6^n} and {(-1)^n}, where C(n):=A000108(n).
Showing 1-2 of 2 results.

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