A243663 Triangle read by rows: the reversed x = 1+q Narayana triangle at m=3.

1, 4, 1, 22, 11, 1, 140, 105, 21, 1, 969, 969, 306, 34, 1, 7084, 8855, 3850, 700, 50, 1, 53820, 80730, 44850, 11500, 1380, 69, 1, 420732, 736281, 498771, 166257, 28665, 2457, 91, 1, 3362260, 6724520, 5379616, 2215136, 503440, 62930, 4060, 116, 1

Offset: 1

Author

N. J. A. Sloane, Jun 13 2014

Keywords

• nonn
• tabl

Comments

See Novelli-Thibon (2014) for precise definition.

Examples

			Triangle begins:
     1
     4,    1
    22,   11,    1
   140,  105,   21,   1
   969,  969,  306,  34,  1
  7084, 8855, 3850, 700, 50, 1
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325
• Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
• J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014. See Fig. 11.

Crossrefs

Cf. A001263, A243662 (m=2).

Programs

Mathematica

T[m_][n_, k_] := Binomial[(m + 1) n + 1 - k, n - k] Binomial[n, k - 1]/n;
Table[T[3][n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 12 2019 *)

Formula

T(n,k) = binomial(4*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n, more generally: T_m(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n and some fixed integer m > 1. - Werner Schulte, Nov 22 2018

Extensions

More terms from Werner Schulte, Nov 22 2018