login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A135356 Triangle T(p,s) read by rows: coefficients in the recurrence of sequences which equal their p-th differences. 11
2, 2, 0, 3, -3, 2, 4, -6, 4, 0, 5, -10, 10, -5, 2, 6, -15, 20, -15, 6, 0, 7, -21, 35, -35, 21, -7, 2, 8, -28, 56, -70, 56, -28, 8, 0, 9, -36, 84, -126, 126, -84, 36, -9, 2, 10, -45, 120, -210, 252, -210, 120, -45, 10, 0, 11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Sequences which equal their p-th differences obey recurrences a(n)=sum(s=1..p) T(p,s)*a(n-s).

This defines T(p,s) as essentially a signed version of a chopped Pascal triangle A014410, see A130785.

For cases like p=2, 4, 6, 8, 10, 12, 14, the denominator of the rational generating function of a(n) contains a factor 1-x; depending on the first terms in the sequences a(n), additional, simpler recurrences may exist if this cancels with a factor in the numerator. - R. J. Mathar, Jun 10 2008

Row sums are 2.

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

T(p,s) = (-1)^(s+1)*A007318(p,s), 1<=s<p. T(p,p) = 0 if p even. T(p,p) = 2 if p odd.

EXAMPLE

Triangle begins with row p=1:

2;

2, 0;

3, -3, 2;

4, -6, 4, 0;

5, -10, 10, -5, 2;

Examples of p=1: A000079, of p=2: A131577, of p=3: A131708, A130785, A131562, A057079, of p=4: A000749, A038503, A009545, A038505, of p=5: A133476, of p=6: A140343, of p=7: A140342.

MAPLE

T:= (p, s)->  `if`(p=s, 2*irem(p, 2), (-1)^(s+1) *binomial(p, s)):

seq(seq(T(p, s), s=1..p), p=1..11);  # Alois P. Heinz, Aug 26 2011

MATHEMATICA

T[p_, s_] := If[p == s, 2*Mod[s, 2], (-1)^(s+1)*Binomial[p, s]]; Table[T[p, s], {p, 1, 11}, {s, 1, p}] // Flatten (* Jean-Fran├žois Alcover, Feb 19 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A130785.

Sequence in context: A095731 A048142 A071426 * A259016 A216504 A216673

Adjacent sequences:  A135353 A135354 A135355 * A135357 A135358 A135359

KEYWORD

sign,tabl

AUTHOR

Paul Curtz, Dec 08 2007, Mar 25 2008, Apr 28 2008

EXTENSIONS

Edited by R. J. Mathar, Jun 10 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 8 04:27 EST 2016. Contains 278902 sequences.