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A352682 Array read by ascending antidiagonals. A(n, k) = (n-1)*Gould(k-1) + Bell(k) for n >= 0 and k >= 1, A(n, 0) = 1.

%I A352682 #20 Apr 15 2024 13:13:06
%S A352682 1,1,0,1,1,1,1,2,2,2,1,3,3,5,6,1,4,4,8,15,21,1,5,5,11,24,52,82,1,6,6,
%T A352682 14,33,83,203,354,1,7,7,17,42,114,324,877,1671,1,8,8,20,51,145,445,
%U A352682 1400,4140,8536,1,9,9,23,60,176,566,1923,6609,21147,46814
%N A352682 Array read by ascending antidiagonals. A(n, k) = (n-1)*Gould(k-1) + Bell(k) for n >= 0 and k >= 1, A(n, 0) = 1.
%C A352682 The array defines a family of Bell-like sequences. The case n = 1 are the Bell numbers A000110, case n = 0 is A032347 and case n = 2 is A038561. The n-th sequence r(k) = T(n, k) is defined for k >= 0 by the recurrence r(k) = Sum_{j=0..k-1} binomial(k-1, j)*r(j) with r(0) = 1 and r(1) = n.
%F A352682 Given a list T let PS(T) denote the list of partial sums of T. Given two list S and T let [S, T] denote the concatenation of the lists. Further let P[end] denote the last element of the list P. Row n of the array with length k can be computed by the following procedure:
%F A352682      A = [n], P = [1], R = [1];
%F A352682      Repeat k-1 times: R = [R, A], P = PS([A, P]), A = [P[end]];
%F A352682      Return R.
%e A352682 Array starts:
%e A352682 n\k 0, 1,  2,  3,  4,   5,    6,    7,     8,      9, ...
%e A352682 ---------------------------------------------------------
%e A352682 [0] 1, 0,  1,  2,  6,  21,   82,  354,  1671,   8536, ... A032347
%e A352682 [1] 1, 1,  2,  5, 15,  52,  203,  877,  4140,  21147, ... A000110
%e A352682 [2] 1, 2,  3,  8, 24,  83,  324, 1400,  6609,  33758, ... A038561
%e A352682 [3] 1, 3,  4, 11, 33, 114,  445, 1923,  9078,  46369, ... A038559
%e A352682 [4] 1, 4,  5, 14, 42, 145,  566, 2446, 11547,  58980, ... A352683
%e A352682 [5] 1, 5,  6, 17, 51, 176,  687, 2969, 14016,  71591, ...
%e A352682 [6] 1, 6,  7, 20, 60, 207,  808, 3492, 16485,  84202, ...
%e A352682 [7] 1, 7,  8, 23, 69, 238,  929, 4015, 18954,  96813, ...
%e A352682 [8] 1, 8,  9, 26, 78, 269, 1050, 4538, 21423, 109424, ...
%e A352682 [9] 1, 9, 10, 29, 87, 300, 1171, 5061, 23892, 122035, ...
%p A352682 alias(PS = ListTools:-PartialSums):
%p A352682 BellRow := proc(n, len) local a, k, P, T;
%p A352682 a := n; P := [1]; T := [1];
%p A352682 for k from 1 to len-1 do
%p A352682    T := [op(T), a]; P := PS([a, op(P)]); a := P[-1] od;
%p A352682 T end: seq(lprint(BellRow(n, 10)), n = 0..9);
%t A352682 nmax = 10;
%t A352682 BellRow[n_, len_] := Module[{a, k, P, T}, a = n; P = {1}; T = {1};
%t A352682    For[k = 1, k <= len - 1, k++,
%t A352682       T = Append[T, a]; P = Accumulate[Join[{a}, P]]; a = P[[-1]]];
%t A352682    T];
%t A352682 rows = Table[BellRow[n, nmax + 1], {n, 0, nmax}];
%t A352682 A[n_, k_] := rows[[n + 1, k + 1]];
%t A352682 Table[A[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 15 2024, after _Peter Luschny_ *)
%o A352682 (Julia)
%o A352682 function BellRow(m, len)
%o A352682     a = m; P = BigInt[1]; T = BigInt[1]
%o A352682     for n in 1:len
%o A352682         T = vcat(T, a)
%o A352682         P = cumsum(vcat(a, P))
%o A352682         a = P[end]
%o A352682     end
%o A352682 T end
%o A352682 for n in 0:9 BellRow(n, 9) |> println end
%Y A352682 Rows: A032347, A000110 (Bell), A038561, A038559, A352683.
%Y A352682 Diagonals: A352684 (main).
%Y A352682 Cf. A040027 (Gould), A352686 (subtriangle).
%Y A352682 Compare A352680 for a similar array based on the Catalan numbers.
%K A352682 nonn,tabl
%O A352682 0,8
%A A352682 _Peter Luschny_, Mar 28 2022