A155998 Triangle read by rows: T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 - (-1)^k)/2.

0, 1, 1, 0, 4, 0, 1, 3, 3, 1, 0, 8, 0, 8, 0, 1, 5, 10, 10, 5, 1, 0, 12, 0, 40, 0, 12, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 16, 0, 112, 0, 112, 0, 16, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 20, 0, 240, 0, 504, 0, 240, 0, 20, 0

Offset: 0

Author

Roger L. Bagula, Feb 01 2009

Keywords

• nonn
• tabl

Comments

Row sums are: A155559(n) = {0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...}.

Examples

			Triangle begins as:
  0;
  1,  1;
  0,  4,  0;
  1,  3,  3,   1;
  0,  8,  0,   8,   0;
  1,  5, 10,  10,   5,   1;
  0, 12,  0,  40,   0,  12,  0;
  1,  7, 21,  35,  35,  21,  7,   1;
  0, 16,  0, 112,   0, 112,  0,  16, 0;
  1,  9, 36,  84, 126, 126, 84,  36, 9,  1;
  0, 20,  0, 240,   0, 504,  0, 240, 0, 20, 0;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Programs

GAP

Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*(2 - (-1)^k*(1 + (-1)^n))/2 ))); # G. C. Greubel, Dec 01 2019

Magma

[Binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2: k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 01 2019

Maple

seq(seq( binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2, k=0..n), n=0..12); # G. C. Greubel, Dec 01 2019

Mathematica

f[n_, k_]:= Binomial[n, k]*(1 - (-1)^k)/2; Table[f[n,k]+f[n,n-k], {n, 0, 10}, {k, 0, n}]//Flatten
Table[Binomial[n, k]*(2-(-1)^k*(1+(-1)^n))/2, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 01 2019 *)

PARI

T(n,k) = binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2; \\ G. C. Greubel, Dec 01 2019

Sage

[[binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Dec 01 2019

Formula

T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 - (-1)^k)/2.

From G. C. Greubel, Dec 01 2019: (Start)

T(n, k) = binomial(n, k)*(2 - (-1)^k*(1 + (-1)^n))/2.

Sum_{k=0..n} T(n,k) = 2^n = A155559(n) for n >= 1.

Sum_{k=0..n-1} T(n,k) = (2^(n+1) - (1-(-1)^n))/2 = A051049(n), n >= 1. (End)