0,8

Row sums are:

{1, 0, 1, 12, 193, 3980, 100805, 3034920, 105994833, 4215106728, 188097696345,...}.

Plotting this function gives what appears to be a von Koch dust set:

a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 243}];

b = Table[If[m <= n, 3 - Mod[a[[n]][[m]], 3], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];

ListDensityPlot[b, Mesh -> False, Frame -> False, AspectRatio -> Automatic, ColorFunction -> (Hue[2# ] &)]

Derivation:

Let:(as permutations)

f(n)/f(m)=(-1)^(n-m)*StrilingS1[n,m];

and assume:

f(m)=n!/((-1)^(n-m)*StrilingS1[n,m]);

Then:f(n-m)=n!/((-1)^(m)*StrilingS1[n,n-m]); Combinations are:

c(n,m)=StirlingS1[n,m]/(n!/((-1)^(m)*StrilingS1[n,n-m]));

Leaving out the n! you get this set.

t(n,m)=((-1)^(n - m)*StirlingS1[n, m]*(-1)^(-m)*StirlingS1[n, n - m]).

{1},

{0, 0},

{0, 1, 0},

{0, 6, 6, 0},

{0, 36, 121, 36, 0},

{0, 240, 1750, 1750, 240, 0},

{0, 1800, 23290, 50625, 23290, 1800, 0},

{0, 15120, 308700, 1193640, 1193640, 308700, 15120, 0},

{0, 141120, 4207896, 25738720, 45819361, 25738720, 4207896, 141120, 0},

{0, 1451520, 59832864, 535810464, 1510458516, 1510458516, 535810464, 59832864, 1451520, 0},

{0, 16329600, 893121120, 11082015000, 45789404640, 72535955625, 45789404640, 11082015000, 893121120, 16329600, 0}

Clear[t, n, m, a];

t[n_, m_] = ((-1)^(n - m)*StirlingS1[n, m]*(-1)^(-m)*StirlingS1[n, n - m]);

Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];

Flatten[%]

Sequence in context: A006806 A002892 A055667 this_sequence A005597 A081825 A028969

Adjacent sequences: A155739 A155740 A155741 this_sequence A155743 A155744 A155745

nonn,tabl,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 26 2009