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 A375550 #22 Feb 08 2025 16:04:50
%S A375550 1,6,1,25,7,1,88,32,8,1,280,120,40,9,1,832,400,160,49,10,1,2352,1232,
%T A375550 560,209,59,11,1,6400,3584,1792,769,268,70,12,1,16896,9984,5376,2561,
%U A375550 1037,338,82,13,1,43520,26880,15360,7937,3598,1375,420,95,14,1
%N A375550 Triangle read by rows: T(m, n, k) = binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1) for m = 4.
%C A375550 Triangle T(m,n,k) is a Riordan array of the form ((1-x)^(m-1)*(1-2x)^(-m-1), x/(1-x)), for m = 3. - _Igor Victorovich Statsenko_, Feb 08 2025
%F A375550 T(m, n, k) = Sum_{j=0..n-k} binomial(m + j, m)*binomial(n + 1, n - (j + k)) for m = 3.
%F A375550 G.f. of column k: (1 - x)^(2 - k) / (1 - 2*x)^4.
%e A375550 Triangle starts:
%e A375550 [0] 1;
%e A375550 [1] 6, 1;
%e A375550 [2] 25, 7, 1;
%e A375550 [3] 88, 32, 8, 1;
%e A375550 [4] 280, 120, 40, 9, 1;
%e A375550 [5] 832, 400, 160, 49, 10, 1;
%e A375550 [6] 2352, 1232, 560, 209, 59, 11, 1;
%e A375550 [7] 6400, 3584, 1792, 769, 268, 70, 12, 1;
%e A375550 [8] 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1;
%e A375550 [9] 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1;
%e A375550 ...
%e A375550 Seen as an array of the columns:
%e A375550 [0] 1, 6, 25, 88, 280, 832, 2352, 6400, 16896, ...
%e A375550 [1] 1, 7, 32, 120, 400, 1232, 3584, 9984, 26880, ...
%e A375550 [2] 1, 8, 40, 160, 560, 1792, 5376, 15360, 42240, ...
%e A375550 [3] 1, 9, 49, 209, 769, 2561, 7937, 23297, 65537, ...
%e A375550 [4] 1, 10, 59, 268, 1037, 3598, 11535, 34832, 100369, ...
%e A375550 [5] 1, 11, 70, 338, 1375, 4973, 16508, 51340, 151709, ...
%e A375550 [6] 1, 12, 82, 420, 1795, 6768, 23276, 74616, 226325, ...
%p A375550 T := (m, n, k) -> binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1);
%p A375550 for n from 0 to 9 do seq(simplify(T(4, n, k)), k = 0..n) od;
%p A375550 # As a binomial sum:
%p A375550 T := (m, n, k) -> add(binomial(m + j, m)*binomial(n + 1, n - (j + k)), j = 0..n-k):
%p A375550 for n from 0 to 9 do [n], seq(T(3, n, k), k = 0..n) od;
%p A375550 # Alternative, generating the array of the columns:
%p A375550 cgf := k -> (1 - x)^(2 - k) / (1 - 2*x)^4:
%p A375550 ser := (k, len) -> series(cgf(k), x, len + 2):
%p A375550 Tcol := (k, len) -> seq(coeff(ser(k, len), x, j), j = 0..len):
%p A375550 seq(lprint([k], Tcol(k, 8)), k = 0..6);
%Y A375550 Column k: A055585 (k=0), A001794 (k=1), A001789 (k=2), A027608 (k=3), A055586 (k=4).
%Y A375550 Cf. A145018 (diagonal n-2), A375549 (row sums), A049612 (alternating row sums), A122433.
%K A375550 nonn,tabl
%O A375550 0,2
%A A375550 _Peter Luschny_, Sep 23 2024