A274293 Triangle read by rows: T(n,k) (0 <= k <= n) given by T(n,0) = 1, T(n,1) = 2^n - 1, T(n,2) = 2^n - 2, T(n,n-1) = T(n,n) = binomial(2n-2,n-1); and the other internal entries satisfy T(n,k) = T(n,k-1) + T(n-1,k).

1, 1, 1, 1, 3, 2, 1, 7, 6, 6, 1, 15, 14, 20, 20, 1, 31, 30, 50, 70, 70, 1, 63, 62, 112, 182, 252, 252, 1, 127, 126, 238, 420, 672, 924, 924, 1, 255, 254, 492, 912, 1584, 2508, 3432, 3432, 1, 511, 510, 1002, 1914, 3498, 6006, 9438, 12870, 12870, 1, 1023, 1022, 2024, 3938, 7436, 13442, 22880, 35750, 48620, 48620

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Niket Gowravaram, A Variation of the nil-Temperley-Lieb algebras of type A, Preprint, 2015.

Example

Triangle begins:
1,
1,1,
1,3,2,
1,7,6,6,
1,15,14,20,20,
1,31,30,50,70,70,
1,63,62,112,182,252,252,
...

Mathematica

T[n_, k_] := T[n, k] = Which[k==0, 1, k<=2, 2^n-k, k==n || k == n-1, Binomial[2*n - 2, n-1], True, T[n, k-1] + T[n-1, k]]; Flatten[ Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Giovanni Resta, Jun 19 2016 *)

Prog

(PARI) T(n, k) = if (k==0, 1, if (k==1, 2^n-1, if (k==2, 2^n-2, if ((k==(n-1)) || (k==n), binomial(2*n-2, n-1), T(n, k-1) + T(n-1, k)))));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", "); ); print(); ); } \\ Michel Marcus, Jun 18 2016

Crossrefs

Cf. A039597, A274292.

Sequence in context: A350571 A277919 A094531 * A161009 A111960 A130462

Adjacent sequences: A274290 A274291 A274292 * A274294 A274295 A274296

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Jun 18 2016

Extensions

More terms from Michel Marcus, Jun 18 2016

Status

approved