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 A292795 #22 Jan 06 2021 19:46:16
%S A292795 1,1,0,1,1,0,1,1,1,0,1,1,3,2,0,1,1,3,7,2,0,1,1,3,13,18,3,0,1,1,3,13,
%T A292795 36,42,4,0,1,1,3,13,60,122,110,5,0,1,1,3,13,60,206,433,250,6,0,1,1,3,
%U A292795 13,60,326,865,1356,627,8,0,1,1,3,13,60,326,1345,3408,4449,1439,10,0
%N A292795 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A292795 Alois P. Heinz, <a href="/A292795/b292795.txt">Antidiagonals n = 0..140, flattened</a>
%F A292795 G.f. of column k: Product_{j>=1} (1+x^j)^A226873(j,k).
%F A292795 A(n,k) = Sum_{j=0..n} A319498(n,j).
%e A292795 A(2,3) = 3: {aa}, {ab}, {ba}.
%e A292795 A(3,2) = 7: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}.
%e A292795 A(3,3) = 13: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}.
%e A292795 Square array A(n,k) begins:
%e A292795 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A292795 0, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A292795 0, 1, 3, 3, 3, 3, 3, 3, 3, ...
%e A292795 0, 2, 7, 13, 13, 13, 13, 13, 13, ...
%e A292795 0, 2, 18, 36, 60, 60, 60, 60, 60, ...
%e A292795 0, 3, 42, 122, 206, 326, 326, 326, 326, ...
%e A292795 0, 4, 110, 433, 865, 1345, 2065, 2065, 2065, ...
%e A292795 0, 5, 250, 1356, 3408, 6228, 9468, 14508, 14508, ...
%e A292795 0, 6, 627, 4449, 15025, 29845, 51325, 76525, 116845, ...
%p A292795 b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
%p A292795 add(b(n-j, j, t-1)/j!, j=i..n/t))
%p A292795 end:
%p A292795 g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
%p A292795 h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A292795 add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
%p A292795 end:
%p A292795 A:= (n, k)-> h(n$2, min(n, k)):
%p A292795 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A292795 b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
%t A292795 g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
%t A292795 h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[h[n - i*j, i - 1, k]*Binomial[g[i, k], j], {j, 0, n/i}]]];
%t A292795 A[n_, k_] := h[n, n, Min[n, k]];
%t A292795 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten(* _Jean-François Alcover_, Jan 02 2021, after _Alois P. Heinz_ *)
%Y A292795 Columns k=0-10 give: A000007, A000009, A340409, A340410, A340411, A340412, A340413, A340414, A340415, A340416, A340417.
%Y A292795 Rows n=0-1 give: A000012, A057427.
%Y A292795 Main diagonal gives A292796.
%Y A292795 Cf. A226873, A292712, A292804, A319498.
%K A292795 nonn,tabl
%O A292795 0,13
%A A292795 _Alois P. Heinz_, Sep 23 2017