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%I #2 Mar 30 2012 17:34:28
%S 1,1,1,1,4,1,1,11,11,1,1,4,110,4,1,1,5,354,354,5,1,1,6,36,4954,36,6,1,
%T 1,7,49,20103,20103,49,7,1,1,8,64,512,361710,512,64,8,1,1,9,81,729,
%U 1813580,1813580,729,81,9,1,1,10,100,1000,10000,39894578,10000,1000,100
%N A vector recursion designed around a factorial row sum : v(n)=if[odd,{1.n,n^2,...,(n+1)!/2-Sum[2^m,{m,0,n/2-1}],(n+1)!/2-Sum2^m,{m,0,n/2-1}],...n^2.n,1}],if[ even{1.n,n^2,...,(n+1)!-2Sum[2^m,{m,0,n/2-1}],...n^2.n,1}].
%C Row sums are:
%C {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800,...}.
%C This designed symmetrical triangle is meant to be like the Eulerian numbers
%C in row sum ( the Stirling numbers of the first kind also have factorial row sums).
%F v(n)=if[odd,{1.n,n^2,...,(n+1)!/2-Sum[2^m,{m,0,n/2-1}],(n+1)!/2-Sum2^m,{m,0,n/2-1}],...n^2.n,1}],
%F if[ even{1.n,n^2,...,(n+1)!-2Sum[2^m,{m,0,n/2-1}],...n^2.n,1}].
%e {1},
%e {1, 1},
%e {1, 4, 1},
%e {1, 11, 11, 1},
%e {1, 4, 110, 4, 1},
%e {1, 5, 354, 354, 5, 1},
%e {1, 6, 36, 4954, 36, 6, 1},
%e {1, 7, 49, 20103, 20103, 49, 7, 1},
%e {1, 8, 64, 512, 361710, 512, 64, 8, 1},
%e {1, 9, 81, 729, 1813580, 1813580, 729, 81, 9, 1},
%e {1, 10, 100, 1000, 10000, 39894578, 10000, 1000, 100, 10, 1}
%t Clear[v, n]; v[0] = {1}; v[1] = {1, 1};
%t v[n_] := v[n] = If[Mod[n, 2] == 0, Join[Table[ n^m, {m,0, Floor[n/2] - 1}], {(n+1)! - 2*Sum[ n^m, {m, 0, Floor[n/2] - 1}]}, Table[ n^m, {m, Floor[n/2] - 1, 0, -1}]],
%t Join[Table[ n^m, {m, 0, Floor[n/2] - 1}], {(n+1)!/2 - Sum[ n^m, {m, 0,Floor[n/2] - 1}], (n+1)!/2 - Sum[ n^m, {m, 0, Floor[n/2] - 1}]}, Table[ n^m, {m, Floor[n/2] - 1, 0, -1}]]]'
%t Table[v[n], {n, 0, 10}];
%t Flatten[%]
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Dec 15 2008
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