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%I #20 Dec 27 2013 07:52:58
%S 0,0,1,0,1,3,0,1,4,12,0,1,6,18,60,0,1,10,32,96,360,0,1,18,66,186,600,
%T 2520,0,1,34,152,426,1222,4320,20160,0,1,66,378,1110,2964,9086,35280,
%U 181440,0,1,130,992,3186,8254,22818,75882,322560,1814400
%N Total sum A(n,k) of the k-th powers of lengths of ascending runs in all permutations of [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A229001/b229001.txt">Rows n = 0..140, flattened</a>
%F A(n,k) = Sum_{t=1..n} t^k * A122843(n,t).
%F For fixed k, A(n,k) ~ n! * n * sum(t>=1, t^k*(t^2+t-1)/(t+2)!) = n! * n * ((Bell(k) - Bell(k+1) + sum(j=0..k, (-1)^j*(2^j*((2*k-j+1)/(j+1))-1) *Bell(k-j)*C(k,j)))*exp(1) - (-1)^k*(2^k-1)), where Bell(k) are Bell numbers A000110. - _Vaclav Kotesovec_, Sep 12 2013
%e A(3,2) = 32 = 9+5+5+5+5+3 = 3^2+4*(2^2+1^2)+3*1^2: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
%e Square array A(n,k) begins:
%e : 0, 0, 0, 0, 0, 0, 0, ...
%e : 1, 1, 1, 1, 1, 1, 1, ...
%e : 3, 4, 6, 10, 18, 34, 66, ...
%e : 12, 18, 32, 66, 152, 378, 992, ...
%e : 60, 96, 186, 426, 1110, 3186, 9846, ...
%e : 360, 600, 1222, 2964, 8254, 25620, 86782, ...
%e : 2520, 4320, 9086, 22818, 66050, 214410, 765506, ...
%p A:= (n, k)-> add(`if`(n=t, 1, n!/(t+1)!*(t*(n-t+1)+1
%p -((t+1)*(n-t)+1)/(t+2)))*t^k, t=1..n):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A[n_, k_] := Sum[If[n == t, 1, n!/(t + 1)!*(t*(n - t + 1) + 1 - ((t + 1)*(n - t) + 1)/(t + 2))]* t^k, {t, 1, n}]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y Columns k=0-10 give: A001710(n+1) for n>0, A001563, A228959, A229003, A228994, A228995, A228996, A228997, A228998, A228999, A229000.
%Y Rows n=0-2 give: A000004, A000012, A052548.
%Y Main diagonal gives: A229002.
%K nonn,tabl
%O 0,6
%A _Alois P. Heinz_, Sep 10 2013
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