|
%I #39 May 04 2023 09:51:59
%S 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,
%T 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,
%U 0,0,132,1485,1925,616,54,1
%N Triangle T(n,k) (n >= 2, k >= 1) of Raney numbers read by rows.
%C There are only m nonzero entries in the m-th column.
%C Related to A033282: shift row n of A033282 triangle n places to the right and transpose the resulting table. - _Michel Marcus_, Feb 04 2014
%H G. C. Greubel, <a href="/A079508/b079508.txt">Rows n=2..100 of triangle, flattened</a>
%H Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Getachew/get3.html">Pattern-Avoidance and Fuss-Catalan Numbers</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
%H G. Kreweras, <a href="http://archive.numdam.org/article/MSH_1976__53__5_0.pdf">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30 (see definition p. 26 and table p. 27).
%H G. N. Raney, <a href="http://www.jstor.org/stable/1993433">Functional composition patterns and power series reversion</a>, Trans. Amer. Math. Soc., 94 (1960), pp. 441-451.
%F T(n,k) = C(k, n-k) * C(n, k+1)/k. - _Michel Marcus_, Feb 04 2014
%e From _Michel Marcus_, Feb 04 2014: (Start)
%e Triangle starts:
%e 1;
%e 0, 1;
%e 0, 2, 1;
%e 0, 0, 5, 1;
%e 0, 0, 5, 9, 1;
%e 0, 0, 0, 21, 14, 1;
%e 0, 0, 0, 14, 56, 20, 1;
%e 0, 0, 0, 0, 84, 120, 27, 1;
%e 0, 0, 0, 0, 42, 300, 225, 35, 1;
%e 0, 0, 0, 0, 0, 330, 825, 385, 44, 1;
%e 0, 0, 0, 0, 0, 132, 1485, 1925, 616, 54, 1;
%e ... (End)
%t Table[Binomial[k, n-k]*Binomial[n, k+1]/k, {n,2,10}, {k,1,n-1}]//Flatten (* _G. C. Greubel_, Jan 17 2019 *)
%o (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
%o (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
%o (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
%o (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
%Y Sum of nonzero entries in each column gives A001003. Alternating sum of each column is 1. Second diagonal on right gives A000096.
%Y Leftmost diagonal is A000108.
%K nonn,tabl
%O 2,5
%A _N. J. A. Sloane_, Jan 21 2003
%E Corrected and extended by _Michel Marcus_, Feb 04 2014
|
