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 A142468 #32 Feb 19 2025 17:31:57
%S A142468 1,1,1,1,9,1,1,45,45,1,1,165,825,165,1,1,495,9075,9075,495,1,1,1287,
%T A142468 70785,259545,70785,1287,1,1,3003,429429,4723719,4723719,429429,3003,
%U A142468 1,1,6435,2147145,61408347,184225041,61408347,2147145,6435,1
%N A142468 An eight-products triangle sequence of coefficients: T(n,k) = binomial(n,k) * Product_{j=1..7} j!*(n+j)!/((k+j)!*(n-k+j)!).
%C A142468 Triangle of generalized binomial coefficients (n,k)_8; cf. A342889. - _N. J. A. Sloane_, Apr 03 2021
%C A142468 Row sums are {1, 2, 11, 92, 1157, 19142, 403691, 10312304, 311348897, 10826298914, 426196716090, ...}.
%C A142468 The general function is T(n,m)_L = binomial(n,m)*Product_{k=1..L} k!*(n + k)!/((m + k)!*(n - m + k)!) to give the quadratic row {1, L+2, 1}.
%H A142468 G. C. Greubel, <a href="/A142468/b142468.txt">Rows n = 0..100 of triangle, flattened</a>
%H A142468 Johann Cigler, <a href="https://arxiv.org/abs/2103.01652">Pascal triangle, Hoggatt matrices, and analogous constructions</a>, arXiv:2103.01652 [math.CO], 2021.
%H A142468 Johann Cigler, <a href="https://www.researchgate.net/publication/349376205_Some_observations_about_Hoggatt_triangles">Some observations about Hoggatt triangles</a>, Universität Wien (Austria, 2021).
%F A142468 T(n,k) = binomial(n,k)*Product_{j=1..7} j!*(n+j)!/((k+j)!*(n-k+j)!).
%e A142468 Triangle begins as:
%e A142468 1;
%e A142468 1, 1;
%e A142468 1, 9, 1;
%e A142468 1, 45, 45, 1;
%e A142468 1, 165, 825, 165, 1;
%e A142468 1, 495, 9075, 9075, 495, 1;
%e A142468 1, 1287, 70785, 259545, 70785, 1287, 1;
%e A142468 1, 3003, 429429, 4723719, 4723719, 429429, 3003, 1;
%e A142468 1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 1;
%p A142468 b:= binomial;
%p A142468 T:= (n, k) -> b(n, k)*mul(b(n+2*j, k+j)/b(n+2*j, j), j = 1..7);
%p A142468 seq(seq(T(n, k), k = 0..n), n = 0..10); # _G. C. Greubel_, Nov 14 2019, Mar 03 2021
%t A142468 T[n_, k_]:= T[n,k]= With[{B=Binomial}, B[n,k]* Product[B[n+2*j,k+j]/B[n+2*j,j], {j, 7}] ];
%t A142468 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Nov 14 2019, Mar 03 2021 *)
%o A142468 (PARI) T(n,k) = b=binomial; b(n,k)*prod(j=1,7, b(n+ 2*j,k+j)/b(n+2*j,j)); \\ _G. C. Greubel_, Nov 14 2019, Mar 03 2021
%o A142468 (Magma) B:=Binomial; [B(n,k)*(&*[B(n+2*j, k+j)/B(n+2*j, j): j in [1..7]]): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 14 2019, Mar 03 2021
%o A142468 (Sage)
%o A142468 b=binomial;
%o A142468 def T(n, k): return b(n,k)*product(b(n+2*j,k+j)/b(n+2*j,j) for j in (1..7))
%o A142468 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 14 2019, Mar 03 2021
%Y A142468 Cf. A001263, A056939, A056940, A056941.
%Y A142468 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 A142468 nonn,tabl
%O A142468 0,5
%A A142468 _Roger L. Bagula_, Sep 20 2008
%E A142468 Edited by _G. C. Greubel_, Nov 14 2019