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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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11
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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; internal format)
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OFFSET
| 1,1
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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 (mathar(AT)strw.leidenuniv.nl), Jun 10 2008
Row sums are 2.
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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.
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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.
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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
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CROSSREFS
| Cf. A130785.
Sequence in context: A095731 A048142 A071426 * A191362 A137422 A139139
Adjacent sequences: A135353 A135354 A135355 * A135357 A135358 A135359
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KEYWORD
| sign,tabl
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AUTHOR
| Paul Curtz (bpcrtz(AT)free.fr), Dec 08 2007, Mar 25 2008, Apr 28 2008
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EXTENSIONS
| Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 10 2008
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