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A342889 Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_10 (n >= 0, 0 <= k <= n).

%I A342889 #53 Feb 06 2025 23:44:42
%S A342889 1,1,1,1,11,1,1,66,66,1,1,286,1716,286,1,1,1001,26026,26026,1001,1,1,
%T A342889 3003,273273,1184183,273273,3003,1,1,8008,2186184,33157124,33157124,
%U A342889 2186184,8008,1,1,19448,14158144,644195552,2254684432,644195552,14158144,19448,1
%N A342889 Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_10 (n >= 0, 0 <= k <= n).
%D A342889 Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
%H A342889 Seiichi Manyama, <a href="/A342889/b342889.txt">Rows n = 0..139, flattened</a>
%H A342889 Boris A. Bondarenko, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/pascal.html">Generalized Pascal Triangles and Pyramids</a>, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993.
%H A342889 Richard L. Ollerton, <a href="/A342889/a342889.pdf">Counting i-paths</a>, Slides of talk presented at Thirteenth International Conference on Fibonacci Numbers and Their Applications, University of Patras (Greece), 2008.
%H A342889 Richard L. Ollerton, <a href="/A342889/a342889_1.pdf">Counting i-paths</a>, Background notes for slides of talk presented at Thirteenth International Conference on Fibonacci Numbers and Their Applications, University of Patras (Greece), 2008.
%H A342889 Richard L. Ollerton and Anthony G. Shannon, <a href="https://doi.org/10.1007/978-0-306-48517-6_19">Extensions of generalized binomial coefficients</a>, In: Howard F.T. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht, pp. 187-199.
%H A342889 Richard L. Ollerton and Anthony G. Shannon, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-2/paper43-2-5.pdf">Further properties of generalized binomial coefficient k-extensions</a>, Fibonacci Quarterly 43.2 (2005): 124-129.
%H A342889 Daniel B. Shapiro, <a href="https://arxiv.org/abs/2302.02243">Divisibility Properties of Integer Sequences</a>, arXiv:2302.02243 [math.NT], 2023. See also <a href="http://math.colgate.edu/~integers/x57/x57.pdf">Integers</a>, (2023) Vol. 23, #A57.
%F A342889 The generalized binomial coefficient (n,k)_m = Product_{j=1..k} binomial(n+m-j,m)/binomial(j+m-1,m).
%e A342889 Triangle begins:
%e A342889   [1],
%e A342889   [1, 1],
%e A342889   [1, 11, 1],
%e A342889   [1, 66, 66, 1],
%e A342889   [1, 286, 1716, 286, 1],
%e A342889   [1, 1001, 26026, 26026, 1001, 1],
%e A342889   [1, 3003, 273273, 1184183, 273273, 3003, 1],
%e A342889   [1, 8008, 2186184, 33157124, 33157124, 2186184, 8008, 1],
%e A342889 ...
%p A342889 # Generalized binomial coefficient:
%p A342889 GBC := proc(n,k,m) local a,j;
%p A342889 a := mul((binomial(n+m-j,m)/binomial(j+m-1,m)),j=1..k);
%p A342889 end;
%p A342889 # Returns first M rows of triangle:
%p A342889 GBCT := proc(m,M) local a,b,n,k; global GBC;
%p A342889 a:=[];
%p A342889 for n from 0 to M do
%p A342889   b:=[seq(GBC(n,k,m),k=0..n)];
%p A342889   a:=[op(a),b];
%p A342889 od: a; end;
%p A342889 GBCT(10,12);
%t A342889 f[n_, k_, m_] := Product[Binomial[n + m - j, m]/Binomial[j + m - 1, m], {j, k}]; Table[f[n, k, 10], {n, 0, 8}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Sep 25 2023 *)
%o A342889 (PARI) f(n, k, m) = prod(j=1, k, binomial(n-j+m, m)/binomial(j-1+m, m));
%o A342889 T(n, k) = f(n, k, 10); \\ _Seiichi Manyama_, Apr 02 2021
%Y A342889 Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.
%K A342889 nonn,tabl
%O A342889 0,5
%A A342889 _N. J. A. Sloane_, Apr 01 2021