A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

0, 1, 0, -1, 2, 0, 3, -2, 4, 0, -5, 6, -4, 8, 0, 11, -10, 12, -8, 16, 0, -21, 22, -20, 24, -16, 32, 0, 43, -42, 44, -40, 48, -32, 64, 0, -85, 86, -84, 88, -80, 96, -64, 128, 0, 171, -170, 172, -168, 176, -160, 192, -128, 256, 0, -341, 342, -340, 344, -336, 352, -320, 384, -256, 512, 0

Offset

0,5

Comments

A variant of the triangle A140503, now including the diagonal.

Since the diagonal contains zeros, rows sums are those of A140503.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = T(n-1, k+1) - T(n-1, k). T(0, k) = A001045(k).

T(n, k) = (2^k - 2^n*(-1)^(n+k))/3, for n >= k >= 0. - Jianing Song, Aug 11 2022

From G. C. Greubel, Feb 18 2023: (Start)

T(n, n-1) = A000079(n).

T(2*n, n) = (-1)^(n+1)*A192382(n+1).

T(2*n, n-1) = (-1)^n*A246036(n-1).

T(2*n, n+1) = A083086(n).

T(3*n, n) = -A115489(n).

Sum_{k=0..n} T(n, k) = A052992(n)*[n>0] + 0*[n=0].

Sum_{k=0..n} (-1)^k*T(n, k) = A045883(n).

Sum_{k=0..n} 2^k*T(n, k) = A084175(n).

Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^(n+1)*A109765(n).

Sum_{k=0..n} 3^k*T(n, k) = A091056(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*A097038(n).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^(n+1)*A138495(n). (End)

Example

Triangle begins as:
    0;
    1,   0;
   -1,   2,   0;
    3,  -2,   4,  0;
   -5,   6,  -4,  8,   0;
   11, -10,  12, -8,  16,  0;
  -21,  22, -20, 24, -16, 32,  0;

Maple

A001045:= n -> (2^n-(-1)^n)/3;
A140944:= proc(n, k) if n = 0 then A001045(k); else procname(n-1, k+1)-procname(n-1, k) ; fi; end:
seq(seq(A140944(n, k), k=0..n), n=0..10); # R. J. Mathar, Sep 07 2009

Mathematica

T[0, 0]=0; T[1, 0]= T[0, 1]= 1; T[0, k_]:= T[0, k]= T[0, k-1] + 2*T[0, k-2]; T[n_, n_]=0; T[n_, k_]:= T[n, k] = T[n-1, k+1] - T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2014 *)
Table[2^k*(1-(-2)^(n-k))/3, {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2023 *)

Prog

(PARI) T(n, k) = (2^k - 2^n*(-1)^(n+k))/3 \\ Jianing Song, Aug 11 2022
(Magma) [2^k*(1-(-2)^(n-k))/3: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 18 2023
(SageMath)
def A140944(n, k): return 2^k*(1 - (-2)^(n-k))/3
flatten([[A140944(n, k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 18 2023

Crossrefs

Cf. A000079, A001045, A016131, A045883, A052992, A083086, A084175.

Cf. A091056, A097038, A109765, A115489, A140503, A192382, A246036.

Sequence in context: A208457 A353335 A232343 * A057860 A092915 A063749

Adjacent sequences: A140941 A140942 A140943 * A140945 A140946 A140947

Keyword

sign,tabl,easy

Author

Paul Curtz, Jul 24 2008

Extensions

Edited and extended by R. J. Mathar, Sep 07 2009

Status

approved