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.
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
Examples
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, ... ...
Links
- Alois P. Heinz, Antidiagonals n = 0..60, flattened
Crossrefs
Programs
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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 *)
Formula
A(n,k) = Sum_{j=0..max(0,n-1)} A173018(n,j)^k.