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.

A112711 Partial sums of Catalan numbers A000108 multiplied by powers of -4.

Original entry on oeis.org

1, -3, 29, -291, 3293, -39715, 500957, -6527779, 87188701, -1187355427, 16424527069, -230141827875, 3259720426717, -46595454638883, 671319066305757, -9738441487391523, 142121594825368797, -2085158937761782563, 30737922595101500637, -455043684091275090723
Offset: 0

Views

Author

Wolfdieter Lang, Oct 31 2005

Keywords

Comments

Fifth column (m=4) of triangle A112707 (partial sums of Catalan numbers multiplied by powers of nonpositive numbers).

Crossrefs

Cf. A112710 (m=3).

Formula

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

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