A135356 Triangle T(n,k) read by rows: coefficients in the recurrence of sequences which equal their n-th differences.

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

Offset: 1

Author

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

Keywords

• sign
• tabl

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

Examples

			Triangle begins with row n=1:
  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;
		

Links

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

Crossrefs

Related sequences: A000079 (n=1), A131577 (n=2), (A131708 , A130785, A131562, A057079) (n=3), (A000749, A038503, A009545, A038505) (n=4), A133476 (n=5), A140343 (n=6), A140342 (n=7).

Cf. A000225, A000984, A051049, A130785.

Programs

Magma

A135356:= func< n,k | k eq n select 1-(-1)^n else (-1)^(k+1)*Binomial(n,k) >;
[A135356(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 09 2023

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, 12}, {s, p}]//Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

SageMath

def A135356(n,k):
    if (k==n): return 2*(n%2)
    else: return (-1)^(k+1)*binomial(n,k)
flatten([[A135356(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Apr 09 2023

Formula

T(n,k) = (-1)^(k+1)*A007318(n, k). T(n,n) = 1 - (-1)^n.

Sum_{k=1..n} T(n, k) = 2.

From G. C. Greubel, Apr 09 2023: (Start)

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = 2*A051049(n-1).

Sum_{k=1..n-1} T(n, k) = (1 + (-1)^n).

Sum_{k=1..n-1} (-1)^(k-1)*T(n, k) = A000225(n-1).

T(2*n, n) = (-1)^(n-1)*A000984(n), n >= 1. (End)

Extensions

Edited by R. J. Mathar, Jun 10 2008