A155997 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.

2, 1, 1, 2, 0, 2, 1, 3, 3, 1, 2, 0, 12, 0, 2, 1, 5, 10, 10, 5, 1, 2, 0, 30, 0, 30, 0, 2, 1, 7, 21, 35, 35, 21, 7, 1, 2, 0, 56, 0, 140, 0, 56, 0, 2, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 2, 0, 90, 0, 420, 0, 420, 0, 90, 0, 2, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset

0,1

Comments

Row sums are: {2, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,...}

Links

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

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 for n >= 1.

Sum_{k=0..n-1} T(n,k) = (2^(n+1) - 3 - (-1)^n)/2 = A140253(n), n >= 2. (End)

Example

Triangle begins as:
  2;
  1, 1;
  2, 0,  2;
  1, 3,  3,  1;
  2, 0, 12,  0,   2;
  1, 5, 10, 10,   5,   1;
  2, 0, 30,  0,  30,   0,   2;
  1, 7, 21, 35,  35,  21,   7,  1;
  2, 0, 56,  0, 140,   0,  56,  0,  2;
  1, 9, 36, 84, 126, 126,  84, 36,  9, 1;
  2, 0, 90,  0, 420,   0, 420,  0, 90, 0, 2;

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)^k*Binomial[n, 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 *)

Prog

(PARI) T(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
(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
(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

Crossrefs

Sequence in context: A194529 A055138 A177717 * A326171 A359393 A123223

Adjacent sequences: A155994 A155995 A155996 * A155998 A155999 A156000

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 01 2009

Status

approved