

A135356


Triangle T(p,s) read by rows: coefficients in the recurrence of sequences which equal their pth differences.


12



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
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OFFSET

1,1


COMMENTS

Sequences which equal their pth differences obey recurrences a(n)=sum(s=1..p) T(p,s)*a(ns).
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 1x; 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 (* JeanFrançois 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



