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A079508
Triangle T(n,k) (n >= 2, k >= 1) of Raney numbers read by rows.
2
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
2,5
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
LINKS
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.
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)
EXAMPLE
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)
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 *)
PROG
(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
(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
(SageMath) [[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
(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
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).
Sequence in context: A394542 A034093 A246187 * A057150 A185663 A262125
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jan 21 2003
EXTENSIONS
Corrected and extended by Michel Marcus, Feb 04 2014
STATUS
approved