A143185 Triangle read by rows: T(n, k) = (1 + abs(n-2*k))*binomial(n,k), with T(n, 0) = T(n, n) = 1.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 6, 12, 1, 1, 20, 20, 20, 20, 1, 1, 30, 45, 20, 45, 30, 1, 1, 42, 84, 70, 70, 84, 42, 1, 1, 56, 140, 168, 70, 168, 140, 56, 1, 1, 72, 216, 336, 252, 252, 336, 216, 72, 1, 1, 90, 315, 600, 630, 252, 630, 600, 315, 90, 1

Offset

0,5

Links

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

Formula

T(n, k) = (1 + abs(n-2*k))*binomial(n,k) for 1 <= k <= n-1, with T(n, 0) = T(n, n) = 1.

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

Example

Table begins as:
  1;
  1,  1;
  1,  2,   1;
  1,  6,   6,   1;
  1, 12,   6,  12,   1;
  1, 20,  20,  20,  20,   1;
  1, 30,  45,  20,  45,  30,   1;
  1, 42,  84,  70,  70,  84,  42,   1;
  1, 56, 140, 168,  70, 168, 140,  56,   1;
  1, 72, 216, 336, 252, 252, 336, 216,  72,  1;
  1, 90, 315, 600, 630, 252, 630, 600, 315, 90, 1;

Mathematica

T[n_, k_] = If[k*(n-k)==0, 1, (1 + Abs[n-2*k])*Binomial[n, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
A143185:= func< n, k | k eq 0 or k eq n select 1 else (1+Abs(n-2*k))*Binomial(n, k) >;
[A143185(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 23 2024
(SageMath)
def A143185(n, k): return 1 if (k==0 or k==n) else (1+abs(n-2*k))*binomial(n , k)
flatten([[A143185(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 23 2024

Crossrefs

Sequence in context: A283795 A168641 A255914 * A347675 A157635 A075798

Adjacent sequences: A143182 A143183 A143184 * A143186 A143187 A143188

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 17 2008

Extensions

Edited by G. C. Greubel, Apr 23 2024

Status

approved