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A152937 A vector recursion designed around a factorial row sum : v(n)=if[odd,{1.n,n^2,...,n!-Sum[2^m,{m,0,n/2-1}],n!-Sum2^m,{m,0,n/2-1}],...n^2.n,1}],if[ even{1.n,n^2,...,n!-2Sum[2^m,{m,0,n/2-1}],...n^2.n,1}]. 0
1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 4, 14, 4, 1, 1, 5, 54, 54, 5, 1, 1, 6, 36, 634, 36, 6, 1, 1, 7, 49, 2463, 2463, 49, 7, 1, 1, 8, 64, 512, 39150, 512, 64, 8, 1, 1, 9, 81, 729, 180620, 180620, 729, 81, 9, 1, 1, 10, 100, 1000, 10000, 3606578, 10000, 1000, 100, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Row sums are:

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

This designed symmetrical triangle is meant to be like the Eulerian numbers

in row sum ( the Stirling numbers of the first kind also have factorial row sums).

LINKS

Table of n, a(n) for n=0..65.

FORMULA

v(n)=if[odd,{1.n,n^2,...,n!-Sum[2^m,{m,0,n/2-1}],n!-Sum2^m,{m,0,n/2-1}],...n^2.n,1}],

if[ even{1.n,n^2,...,n!-2Sum[2^m,{m,0,n/2-1}],...n^2.n,1}].

EXAMPLE

{1},

{1, 1},

{1, 0, 1},

{1, 2, 2, 1},

{1, 4, 14, 4, 1},

{1, 5, 54, 54, 5, 1},

{1, 6, 36, 634, 36, 6, 1},

{1, 7, 49, 2463, 2463, 49, 7, 1},

{1, 8, 64, 512, 39150, 512, 64, 8, 1},

{1, 9, 81, 729, 180620, 180620, 729, 81, 9, 1},

{1, 10, 100, 1000, 10000, 3606578, 10000, 1000, 100, 10, 1}

MATHEMATICA

Clear[v, n]; v[0] = {1}; v[1] = {1, 1};

v[n_] := v[n] = If[Mod[n, 2] == 0, Join[Table[ n^m, {m, 0, Floor[n/2] - 1}], {n! - 2*Sum[ n^m, {m, 0, Floor[n/2] - 1}]}, Table[ n^m, {m, Floor[n/2] - 1, 0, -1}]],

Join[Table[ n^m, {m, 0, Floor[n/2] - 1}], {n!/2 - Sum[ n^m, {m, 0, Floor[n/2] - 1}], n!/2 - Sum[ n^m, {m, 0, Floor[n/2] - 1}]}, Table[ n^m, {m, Floor[n/2] - 1, 0, -1}]]]'

Table[v[n], {n, 0, 10}];

Flatten[%]

CROSSREFS

Sequence in context: A176663 A113021 A298261 * A064552 A209543 A178655

Adjacent sequences:  A152934 A152935 A152936 * A152938 A152939 A152940

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Dec 15 2008

STATUS

approved

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Last modified March 26 12:43 EDT 2019. Contains 321497 sequences. (Running on oeis4.)