A243661 Triangle read by rows: the x = 1+q Narayana triangle at m=3.

1, 4, 3, 22, 33, 12, 140, 315, 231, 55, 969, 2907, 3213, 1547, 273, 7084, 26565, 39270, 28560, 10200, 1428, 53820, 242190, 448500, 437000, 235980, 66861, 7752, 420732, 2208843, 4916457, 6009003, 4351347, 1864863, 437437, 43263, 3362260, 20173560, 52451256, 77134200, 70122000, 40320150, 14307150, 2861430, 246675

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,     3;
    22,    33,    12;
   140,   315,   231,    55;
   969,  2907,  3213,  1547,   273;
  7084, 26565, 39270, 28560, 10200,  1428;
  ...
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..210 (The first 20 rows)
• J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 9.

Crossrefs

The left column is A002293, the main diagonal is A001764.

The case m=1 is A126216 or A033282 (its mirror image).

The case m=2 is A243660.

Programs

Mathematica

polrecip[P_, x_] := P /. x -> 1/x // Together // Numerator;
P[n_, m_] := Sum[Binomial[m n + 1, k] Binomial[(m + 1) n - k, n - k] (1 - x)^k x^(n - k), {k, 0, n}]/(m n + 1);
T[m_] := Reap[For[i=1, i <= 20, i++, z = polrecip[P[i, m], x] /. x -> 1+q; Sow[CoefficientList[z, q]]]][[2, 1]];
T[3] // Flatten (* Jean-François Alcover, Oct 08 2018, from PARI *)

PARI

N(n,m)=sum(k=0,n,binomial(m*n+1,k)*binomial((m+1)*n-k,n-k)*(1-x)^k*x^(n-k))/(m*n+1);
T(m)=for(i=1,20,z=subst(polrecip(N(i,m)),x,1+q);print(Vecrev(z)));
T(3) /* Lars Blomberg, Jul 17 2017 */

Formula

From Werner Schulte, Nov 23 2018: (Start)

T(n,k) = binomial(4*n+1-k,n-k) * binomial(3*n,k-1) / n.

More generally: T_m(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(m*n,k-1) / n where m = 3.

Sum_{k=1..n} (-1)^k * T(n,k) = -1. (End)

Sum_{k = 1..n} (-1)^(k+1)*T(n,k)*binomial(x + 4*n - k + 1, 4*n - k + 1) = (x + 1) * ( Product_{k = 2..n} (x + k)^2 ) * ( Product_{k = 1..2*n+1} (x + n + k) ) / (n!*(3*n + 1)!) for n >= 1. Cf. A126216 and A243660. - Peter Bala, Oct 08 2022

Extensions

a(22)-a(39) from Lars Blomberg, Jul 12 2017