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A335545
A(n,k) is the sum of the k-th powers of the (positive) number of permutations of [n] with j descents (j=0..max(0,n-1)); square array A(n,k), n>=0, k>=0, read by antidiagonals.
4
1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 6, 4, 1, 1, 2, 18, 24, 5, 1, 1, 2, 66, 244, 120, 6, 1, 1, 2, 258, 2664, 5710, 720, 7, 1, 1, 2, 1026, 29284, 322650, 188908, 5040, 8, 1, 1, 2, 4098, 322104, 19888690, 55457604, 8702820, 40320, 9, 1, 1, 2, 16386, 3543124, 1276095330, 16657451236, 17484605040, 524888040, 362880, 10
OFFSET
0,6
LINKS
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
Columns k=0-2 give: A028310, A000142, A168562.
Rows n=0+1, 2-3 give: A000012, A007395(k+1), A178789(k+1).
Main diagonal gives A335546.
Sequence in context: A116855 A173265 A157744 * A334997 A030111 A096921
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 12 2020
STATUS
approved