A352686 Triangle read by rows. T(n, k) = (n-1)*Gould(k-1) + Bell(k) for n >= 0 and k >= 1, T(n, 0) = 1.

1, 1, 1, 1, 2, 3, 1, 3, 4, 11, 1, 4, 5, 14, 42, 1, 5, 6, 17, 51, 176, 1, 6, 7, 20, 60, 207, 808, 1, 7, 8, 23, 69, 238, 929, 4015, 1, 8, 9, 26, 78, 269, 1050, 4538, 21423, 1, 9, 10, 29, 87, 300, 1171, 5061, 23892, 122035, 1, 10, 11, 32, 96, 331, 1292, 5584, 26361, 134646, 738424

Offset: 0

Author

Peter Luschny, Mar 31 2022

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 1, 1;
[2] 1, 2,  3;
[3] 1, 3,  4, 11;
[4] 1, 4,  5, 14, 42;
[5] 1, 5,  6, 17, 51, 176;
[6] 1, 6,  7, 20, 60, 207,  808;
[7] 1, 7,  8, 23, 69, 238,  929, 4015;
[8] 1, 8,  9, 26, 78, 269, 1050, 4538, 21423;
[9] 1, 9, 10, 29, 87, 300, 1171, 5061, 23892, 122035;
		

Crossrefs

Subtriangle of A352682. Main diagonal A352684.

Cf. A000110 (Bell), A040027 (Gould).

Programs

Julia

function A352686Row(n)
    a = BigInt(n == 0 ? 1 : n)
    P = BigInt[1]; T = BigInt[1]
    for k in 1:n
        T = push!(T, a)
        P = cumsum(pushfirst!(P, a))
        a = P[end]
    end
T end
for n in 0:9 println(A352686Row(n)) end

Maple

Bell := n -> combinat:-bell(n):
Gould := proc(n) option remember; ifelse(n = 0, 1,
add(binomial(n, k-1)*Gould(n-k), k = 1..n)) end:
T := (n, k) -> (n-1)*Gould(k-1) + Bell(k):
for n from 0 to 9 do seq(T(n,k), k = 0..n) od;
# Alternative:
alias(PS = ListTools:-PartialSums):
A352686Row := proc(n) local a, k, P, R; a := n; P := [1]; R := [1];
for k from 1 to n do R := [op(R), a]; P := PS([a, op(P)]); a := P[-1] od; R end:
seq(print(A352686Row(n)), n = 0..9);

Mathematica

gould[n_] := gould[n] = If[n == 0, 1, Sum[Binomial[n, k+1]*gould[k], {k, 0, n-1}]];
T[n_, k_] := (n-1) gould[k-1] + BellB[k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2023, after first Maple program *)

Formula

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 triangle T can be computed by the following procedure:

A = [n], P = [1], R = [1];

Repeat n times: R = [R, A], P = PS([A, P]), A = [P[end]];

Return R.