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Search: id:A135356
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| A135356 |
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Triangle T(p,s) read by rows: coefficients in the recurrence of sequences which equal their p-th differences. |
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+0
11
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| 2, 0, 2, 3, -3, 2, 0, 4, -6, 4, 5, -10, 10, -5, 2, 0, 6, -15, 20, -15, 6, 7, -21, 35, -35, 21, -7, 2, 0, 8, -28, 56, -70, 56, -28, 8, 9, -36, 84, -126, 126, -84, 36, -9, 2, 0, 10, -45, 120, -210, 252, -210, 120, -45, 10
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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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 (mathar(AT)strw.leidenuniv.nl), Jun 10 2008
Correction Mar 25 2008: even rows don't begin but end with a 0. Hence triangle is 2; 2, 0; 3, -3, 2; 4, -6, 4, 0; 2, 0, ...
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FORMULA
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T(p,s) = (-1)^(s+1)*A007318(p,s), 1<=s<p. T(p,p)=0 if p even. T(p,p)=0 if p odd.
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EXAMPLE
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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.
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CROSSREFS
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Cf. A130785.
Sequence in context: A033769 A074660 A002125 this_sequence A003987 A063180 A028376
Adjacent sequences: A135353 A135354 A135355 this_sequence A135357 A135358 A135359
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KEYWORD
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sign,tabl
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AUTHOR
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Paul Curtz (bpcrtz(AT)free.fr), Dec 08 2007, Mar 25 2008, Apr 28 2008
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EXTENSIONS
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Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 10 2008
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