A243660 Triangle read by rows: the x = 1+q Narayana triangle at m=2.

1, 3, 2, 12, 16, 5, 55, 110, 70, 14, 273, 728, 702, 288, 42, 1428, 4760, 6160, 3850, 1155, 132, 7752, 31008, 50388, 42432, 19448, 4576, 429, 43263, 201894, 395010, 418950, 259350, 93366, 18018, 1430, 246675, 1315600, 3010700, 3853696, 3010700, 1466080, 433160, 70720, 4862

Offset: 1

Author

N. J. A. Sloane, Jun 13 2014

Keywords

• nonn
• tabl

Comments

See Novelli-Thibon (2014) for precise definition.

The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*...*(x+2n+1) / (n! * (2n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 09 2022

The Maple code T(n,k) := binomial(3*n+1-k,n-k)*binomial(2*n,k-1)/n: with(sumtools): sumrecursion( (-1)^(k+1)*T(n,k)*binomial(x+3*n-k+1, 3*n-k+1), k, s(n) ); returns the recurrence 2*(2*n+1)*n^2*s(n) = (x+n)*(x+2*n)*(x+2*n+1)*s(n-1). The above observation follows from this. - Peter Bala, Oct 30 2022

Examples

			Triangle begins:
     1;
     3,    2;
    12,   16,    5;
    55,  110,   70,   14;
   273,  728,  702,  288,   42;
  1428, 4760, 6160, 3850, 1155,  132;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
• J.-C. Novelli and 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. 8.

Crossrefs

Row sums give A034015(n-1).

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

The case m=3 is A243661.

The right diagonal is A000108.

The left column is A001764.

Same table as A383453, transposed. - Dan Eilers, May 06 2025

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[2] // 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(2)  /* Lars Blomberg, Jul 17 2017 */

PARI

T(n,k) = binomial(3*n+1-k,n-k) * binomial(2*n,k-1) / n; \\ Andrew Howroyd, Nov 23 2018

Formula

From Werner Schulte, Nov 23 2018: (Start)

T(n,k) = binomial(3*n+1-k,n-k) * binomial(2*n,k-1) / n.

More generally: T(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(m*n,k-1) / n, where m = 2.

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

Extensions

Corrected example and a(22)-a(43) from Lars Blomberg, Jul 12 2017

a(44)-a(45) from Werner Schulte, Nov 23 2018