This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A079508 #51 Aug 16 2025 03:18:46
%S A079508 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 A079508 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 A079508 0,0,132,1485,1925,616,54,1,0,0,0,0,0,0,1287,5005,4004,936,65,1
%N A079508 Triangle T(n,k) (n >= 2, k >= 1) of Raney numbers read by rows.
%C A079508 There are only m nonzero entries in the m-th column.
%C A079508 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 A079508 G. C. Greubel, <a href="/A079508/b079508.txt">Rows n=2..100 of triangle, flattened</a>
%H A079508 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 A079508 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 A079508 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 A079508 T(n,k) = binomial(k, n-k) * binomial(n, k+1)/k. - _Michel Marcus_, Feb 04 2014
%F A079508 From _Andrew Howroyd_, Jan 24 2025: (Start)
%F A079508 T(n,k) = A033282(k + 2, n - k - 1) = A126216(k, 2*k - n).
%F A079508 G.f.: -1 + ((1 + y*x) - sqrt(1 - 2*y*x + (y^2 - 4*y)*x^2))/(2*x*y*(1 + x)).
%F A079508 G.f.: -1 + (1/(x*y))*Series_Reversion(x*(1 - x)/(y - y*x + x^2)). (End)
%e A079508 From _Michel Marcus_, Feb 04 2014: (Start)
%e A079508 Triangle starts:
%e A079508 1;
%e A079508 0, 1;
%e A079508 0, 2, 1;
%e A079508 0, 0, 5, 1;
%e A079508 0, 0, 5, 9, 1;
%e A079508 0, 0, 0, 21, 14, 1;
%e A079508 0, 0, 0, 14, 56, 20, 1;
%e A079508 0, 0, 0, 0, 84, 120, 27, 1;
%e A079508 0, 0, 0, 0, 42, 300, 225, 35, 1;
%e A079508 0, 0, 0, 0, 0, 330, 825, 385, 44, 1;
%e A079508 0, 0, 0, 0, 0, 132, 1485, 1925, 616, 54, 1;
%e A079508 ... (End)
%t A079508 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 A079508 (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 A079508 (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 A079508 (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 A079508 (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 A079508 Row sums give A005043.
%Y A079508 Column sums give A001003.
%Y A079508 Alternating sum of each column is 1.
%Y A079508 Second diagonal on right gives A000096.
%Y A079508 Central terms give A000108.
%Y A079508 Cf. A033282, A126216 (transposed variants).
%K A079508 nonn,tabl
%O A079508 2,5
%A A079508 _N. J. A. Sloane_, Jan 21 2003
%E A079508 Corrected and extended by _Michel Marcus_, Feb 04 2014