A079508 Triangle T(n,k) (n >= 2, k >= 1) of Raney numbers read by rows.

1, 0, 1, 0, 2, 1, 0, 0, 5, 1, 0, 0, 5, 9, 1, 0, 0, 0, 21, 14, 1, 0, 0, 0, 14, 56, 20, 1, 0, 0, 0, 0, 84, 120, 27, 1, 0, 0, 0, 0, 42, 300, 225, 35, 1, 0, 0, 0, 0, 0, 330, 825, 385, 44, 1, 0, 0, 0, 0, 0, 132, 1485, 1925, 616, 54, 1, 0, 0, 0, 0, 0, 0, 1287, 5005, 4004, 936, 65, 1

Offset: 2

Author

N. J. A. Sloane, Jan 21 2003

Keywords

• nonn
• tabl

Comments

There are only m nonzero entries in the m-th column.

Related to A033282: shift row n of A033282 triangle n places to the right and transpose the resulting table. - Michel Marcus, Feb 04 2014

Examples

			From _Michel Marcus_, Feb 04 2014: (Start)
Triangle starts:
  1;
  0, 1;
  0, 2, 1;
  0, 0, 5,  1;
  0, 0, 5,  9,  1;
  0, 0, 0, 21, 14,   1;
  0, 0, 0, 14, 56,  20,    1;
  0, 0, 0,  0, 84, 120,   27,    1;
  0, 0, 0,  0, 42, 300,  225,   35,   1;
  0, 0, 0,  0,  0, 330,  825,  385,  44,  1;
  0, 0, 0,  0,  0, 132, 1485, 1925, 616, 54, 1;
  ... (End)
		

Links

• G. C. Greubel, Rows n=2..100 of triangle, flattened
• Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, Pattern-Avoidance and Fuss-Catalan Numbers, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
• G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30 (see definition p. 26 and table p. 27).
• G. N. Raney, Functional composition patterns and power series reversion, Trans. Amer. Math. Soc., 94 (1960), pp. 441-451.

Crossrefs

Row sums give A005043.

Column sums give A001003.

Alternating sum of each column is 1.

Second diagonal on right gives A000096.

Central terms give A000108.

Cf. A033282, A126216 (transposed variants).

Programs

GAP

Flat(List([1..10], n->List([1..n-1], k-> Binomial(k,n-k)*Binomial(n ,k+1)/k ))); # G. C. Greubel, Jan 17 2019

Magma

[[Binomial(k,n-k)*Binomial(n,k+1)/k: k in [1..n-1]]: n in [2..10]]; // G. C. Greubel, Jan 17 2019

Mathematica

Table[Binomial[k, n-k]*Binomial[n, k+1]/k, {n,2,10}, {k,1,n-1}]//Flatten (* G. C. Greubel, Jan 17 2019 *)

PARI

tabl(nn) = {for (n = 2, nn, for (k = 1, n-1, print1(binomial(k, n-k)*binomial(n, k+1)/k, ", ");); print(););} \\ Michel Marcus, Feb 04 2014

Sage

[[binomial(k,n-k)*binomial(n,k+1)/k for k in (1..n-1)] for n in (2..10)] # G. C. Greubel, Jan 17 2019

Formula

T(n,k) = binomial(k, n-k) * binomial(n, k+1)/k. - Michel Marcus, Feb 04 2014

From Andrew Howroyd, Jan 24 2025: (Start)

T(n,k) = A033282(k + 2, n - k - 1) = A126216(k, 2*k - n).

G.f.: -1 + ((1 + y*x) - sqrt(1 - 2*y*x + (y^2 - 4*y)*x^2))/(2*x*y*(1 + x)).

G.f.: -1 + (1/(x*y))*Series_Reversion(x*(1 - x)/(y - y*x + x^2)). (End)

Extensions

Corrected and extended by Michel Marcus, Feb 04 2014