A174345 Triangle T(n, k) = (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(k-1) if floor(n/2) >= k, otherwise (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(n-k), read by rows.

1, 1, 1, 1, 6, 1, 1, 12, 12, 1, 1, 20, 80, 20, 1, 1, 30, 200, 200, 30, 1, 1, 42, 420, 1400, 420, 42, 1, 1, 56, 784, 3920, 3920, 784, 56, 1, 1, 72, 1344, 9408, 28224, 9408, 1344, 72, 1, 1, 90, 2160, 20160, 84672, 84672, 20160, 2160, 90, 1

Offset

1,5

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n, k) = (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(k-1) if floor(n/2) >= k, otherwise (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(n-k).

T(n, n-k) = T(n, k).

From G. C. Greubel, Nov 28 2021: (Start)

T(n, n-1) = A180291(n), n > 1.

T(n, n-1) = 2*A000217(n-1), n > 2. (End)

Example

Triangle begins as:
  1;
  1,  1;
  1,  6,    1;
  1, 12,   12,     1;
  1, 20,   80,    20,     1;
  1, 30,  200,   200,    30,     1;
  1, 42,  420,  1400,   420,    42,     1;
  1, 56,  784,  3920,  3920,   784,    56,    1;
  1, 72, 1344,  9408, 28224,  9408,  1344,   72,  1;
  1, 90, 2160, 20160, 84672, 84672, 20160, 2160, 90,  1;

Mathematica

Table[(Binomial[n-1, k-1]*Binomial[n, k-1]/k)*If[Floor[n/2]>=k, 2^(k-1), 2^(n-k)], {n, 12}, {k, n}]//Flatten

Prog

(Sage)
def A174345(n, k):
    b=binomial
    if ((n//2)>k-1): return (1/(n+1))*b(n-1, k-1)*b(n+1, k)*2^(k-1)
    else: return (1/(n+1))*b(n-1, k-1)*b(n+1, k)*2^(n-k)
flatten([[A174345(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Nov 28 2021

Crossrefs

Cf. A000217, A180291.

Sequence in context: A202868 A202877 A174124 * A174449 A174150 A202673

Adjacent sequences: A174342 A174343 A174344 * A174346 A174347 A174348

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 16 2010

Extensions

Edited by G. C. Greubel, Nov 28 2021

Status

approved