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A vector sequence with set row sum function: row(n)=(n+1)! and linear build up and decline function: f(n,m)=Floor[(m/n)*row(n)].
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%I #3 Mar 30 2012 17:34:28

%S 1,1,1,1,4,1,1,11,11,1,1,15,88,15,1,1,72,287,287,72,1,1,420,840,2518,

%T 840,420,1,1,2880,5760,11519,11519,5760,2880,1,1,22680,45360,68040,

%U 90718,68040,45360,22680,1,1,201600,403200,604800,604799,604799,604800

%N A vector sequence with set row sum function: row(n)=(n+1)! and linear build up and decline function: f(n,m)=Floor[(m/n)*row(n)].

%C row sums (n+1)!:

%C {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800,...}

%F row(n)=(n+1)!: f(n,m)=Floor[(m/n)*row(n)].

%e {1},

%e {1, 1},

%e {1, 4, 1},

%e {1, 11, 11, 1},

%e {1, 15, 88, 15, 1},

%e {1, 72, 287, 287, 72, 1},

%e {1, 420, 840, 2518, 840, 420, 1},

%e {1, 2880, 5760, 11519, 11519, 5760, 2880, 1},

%e {1, 22680, 45360, 68040, 90718, 68040, 45360, 22680, 1},

%e {1, 201600, 403200, 604800, 604799, 604799, 604800, 403200, 201600, 1},

%e {1, 1995840, 3991680, 5987520, 7983360, -2, 7983360, 5987520, 3991680, 1995840, 1}

%t Clear[v, n, row, f]; row[n_] = (n+1);

%t f[n_, m_] = Floor[(m/n)*row[n]/2]; v[0] = {1}; v[1] = {1, 1};

%t v[n_] := v[n] = If[Mod[n, 2] == 0, Join[{1}, Table[ f[n, m], {m, 1, Floor[ n/2] - 1}], {row[n] - 2*Sum[ f[n, m], {m, 1, Floor[n/2] - 1}] - 2}, Table[ f[n, m], {m, Floor[n/ 2] - 1, 1, -1}], { 1}],

%t Join[{1}, Table[ f[n, m], {m, 1, Floor[n/2] - 1}], {row[n]/2 - Sum[ f[n, m], { m, 1, Floor[n/2] - 1}] - 1, row[n]/ 2 - Sum[ f[n, m], {m, 1, Floor[ n/2] - 1}] - 1}, Table[ f[n, m], {m, Floor[n/ 2] - 1, 1, -1}], {1}]];

%t Table[v[n], {n, 0, 10}]; Flatten[%]

%K nonn,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Dec 16 2008