OFFSET
0,6
LINKS
Alois P. Heinz, Antidiagonals n = 0..60, flattened
FORMULA
A(n,k) = Sum_{j=0..max(0,n-1)} A173018(n,j)^k.
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, ...
3, 6, 18, 66, 258, 1026, ...
4, 24, 244, 2664, 29284, 322104, ...
5, 120, 5710, 322650, 19888690, 1276095330, ...
6, 720, 188908, 55457604, 16657451236, 5025377832180, ...
...
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
expand(add(b(u-j, o+j-1, 1)*x^t, j=1..u))+
add(b(u+j-1, o-j, 1), j=1..o))
end:
A:= (n, k)-> (p-> add(coeff(p, x, i)^k, i=0..degree(p)))(b(n, 0$2)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
A:= (n, k)-> add(combinat[eulerian1](n, j)^k, j=0..max(0, n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
B[n_, k_] := B[n, k] = Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
A[0, _] = 1; A[n_, k_] := Sum[B[n, j]^k, {j, 0, n-1}];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)
CROSSREFS
Main diagonal gives A335546.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 12 2020
STATUS
approved