A131562 -id:A131562 - cp's OEIS frontend

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

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

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

			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;

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

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.

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

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

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 *)

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

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)

Edited by R. J. Mathar, Jun 10 2008

A158927 a(n) = -3a(n-1) - 3a(n-2) - 2a(n-3), n > 3.

Original entry on oeis.org

2, 2, 2, -7, 11, -16, 29, -61, 128, -259, 515, -1024, 2045, -4093, 8192, -16387, 32771, -65536, 131069, -262141, 524288, -1048579, 2097155, -4194304, 8388605, -16777213, 33554432, -67108867, 134217731, -268435456, 536870909, -1073741821, 2147483648

Offset: 0

Paul Curtz, Mar 31 2009

sign

The inverse binomial transform of A153130, after dropping A153130(0).
The inverse binomial transform of the full A153130 is A158916.
Dropping two initial terms of A153130 yields A158935, dropping three yields essentially a sign-reversed version of A158916, dropping 4 essentially the sequence here.

Muniru A Asiru, Table of n, a(n) for n = 0..600
Index entries for linear recurrences with constant coefficients, signature (-3, -3, -2).

Same recurrence as A131562, A158916, A158926.

a := [2,2,2,-7];; for n in [5..10^3] do a[n] := -3*a[n-1] - 3*a[n-2] - 2*a[n-3]; od; a; # Muniru A Asiru, Jan 27 2018

a := proc(n) option remember: if n=0 then 2 elif n=1 then 2 elif n=2 then 2 elif n=3 then -7 elif n>=4 then -3*procname(n-1) - 3*procname(n-2) - 2*procname(n-3) fi; end:
seq(a(n), n=0..100); # Muniru A Asiru, Jan 27 2018

a(n) = -3a(n-1) - 3a(n-2) - 2a(n-3), with a(0)=a(1)=a(2)=2, a(3)=-7.
a(n) = (-1)^(n+1)*A157823(n) - A099838(n+3).
G.f.: (2+8*x+14*x^2+9*x^3)/((2*x+1)*(1+x+x^2)). - R. J. Mathar, Apr 09 2009
a(0)=2; a(n) = (1/2)*(-2)^n - 3*cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3) for n >= 1. - Richard Choulet, Apr 23 2009

Edited and extended by R. J. Mathar, Apr 09 2009

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

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