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 A229001 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. 12
 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, 2520, 0, 1, 34, 152, 426, 1222, 4320, 20160, 0, 1, 66, 378, 1110, 2964, 9086, 35280, 181440, 0, 1, 130, 992, 3186, 8254, 22818, 75882, 322560, 1814400 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA A(n,k) = Sum_{t=1..n} t^k * A122843(n,t). 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 EXAMPLE 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). Square array A(n,k) begins: : 0, 0, 0, 0, 0, 0, 0, ... : 1, 1, 1, 1, 1, 1, 1, ... : 3, 4, 6, 10, 18, 34, 66, ... : 12, 18, 32, 66, 152, 378, 992, ... : 60, 96, 186, 426, 1110, 3186, 9846, ... : 360, 600, 1222, 2964, 8254, 25620, 86782, ... : 2520, 4320, 9086, 22818, 66050, 214410, 765506, ... MAPLE A:= (n, k)-> add(`if`(n=t, 1, n!/(t+1)!*(t*(n-t+1)+1 -((t+1)*(n-t)+1)/(t+2)))*t^k, t=1..n): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA 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 *) CROSSREFS Columns k=0-10 give: A001710(n+1) for n>0, A001563, A228959, A229003, A228994, A228995, A228996, A228997, A228998, A228999, A229000. Rows n=0-2 give: A000004, A000012, A052548. Main diagonal gives: A229002. Sequence in context: A213191 A352449 A079520 * A208981 A357892 A261158 Adjacent sequences: A228998 A228999 A229000 * A229002 A229003 A229004 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 10 2013 STATUS approved

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Last modified March 2 04:27 EST 2024. Contains 370460 sequences. (Running on oeis4.)