|
%I #23 Jun 19 2016 07:33:07
%S 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,
%T 252,252,1,127,126,238,420,672,924,924,1,255,254,492,912,1584,2508,
%U 3432,3432,1,511,510,1002,1914,3498,6006,9438,12870,12870,1,1023,1022,2024,3938,7436,13442,22880,35750,48620,48620
%N 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).
%H Niket Gowravaram, <a href="https://math.mit.edu/research/highschool/primes/materials/2015/Niket.pdf">A Variation of the nil-Temperley-Lieb algebras of type A</a>, Preprint, 2015.
%e Triangle begins:
%e 1,
%e 1,1,
%e 1,3,2,
%e 1,7,6,6,
%e 1,15,14,20,20,
%e 1,31,30,50,70,70,
%e 1,63,62,112,182,252,252,
%e ...
%t 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 *)
%o (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)))));
%o tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print(););} \\ _Michel Marcus_, Jun 18 2016
%Y Cf. A039597, A274292.
%K nonn,tabl,easy
%O 0,5
%A _N. J. A. Sloane_, Jun 18 2016
%E More terms from _Michel Marcus_, Jun 18 2016
|
