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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.

%I A335545 #29 Feb 11 2021 11:15:34
%S A335545 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,
%T A335545 258,2664,5710,720,7,1,1,2,1026,29284,322650,188908,5040,8,1,1,2,4098,
%U A335545 322104,19888690,55457604,8702820,40320,9,1,1,2,16386,3543124,1276095330,16657451236,17484605040,524888040,362880,10
%N 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.
%H A335545 Alois P. Heinz, <a href="/A335545/b335545.txt">Antidiagonals n = 0..60, flattened</a>
%F A335545 A(n,k) = Sum_{j=0..max(0,n-1)} A173018(n,j)^k.
%e A335545 Square array A(n,k) begins:
%e A335545   1,   1,      1,        1,           1,             1, ...
%e A335545   1,   1,      1,        1,           1,             1, ...
%e A335545   2,   2,      2,        2,           2,             2, ...
%e A335545   3,   6,     18,       66,         258,          1026, ...
%e A335545   4,  24,    244,     2664,       29284,        322104, ...
%e A335545   5, 120,   5710,   322650,    19888690,    1276095330, ...
%e A335545   6, 720, 188908, 55457604, 16657451236, 5025377832180, ...
%e A335545   ...
%p A335545 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A335545       expand(add(b(u-j, o+j-1, 1)*x^t, j=1..u))+
%p A335545              add(b(u+j-1, o-j, 1), j=1..o))
%p A335545     end:
%p A335545 A:= (n, k)-> (p-> add(coeff(p, x, i)^k, i=0..degree(p)))(b(n, 0$2)):
%p A335545 seq(seq(A(n, d-n), n=0..d), d=0..10);
%p A335545 # second Maple program:
%p A335545 A:= (n, k)-> add(combinat[eulerian1](n, j)^k, j=0..max(0, n-1)):
%p A335545 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A335545 B[n_, k_] := B[n, k] = Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
%t A335545 A[0, _] = 1; A[n_, k_] := Sum[B[n, j]^k, {j, 0, n-1}];
%t A335545 Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 11 2021 *)
%Y A335545 Columns k=0-2 give: A028310, A000142, A168562.
%Y A335545 Rows n=0+1, 2-3 give: A000012, A007395(k+1), A178789(k+1).
%Y A335545 Main diagonal gives A335546.
%Y A335545 Cf. A008292, A173018, A334622.
%K A335545 nonn,tabl
%O A335545 0,6
%A A335545 _Alois P. Heinz_, Sep 12 2020