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A156188 A generalized recursion triangle sequence : m=5; e(n,k,n)=(k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m); t(n,k)=e(n,k,m)+e(n,n-k,m). 0
2, 1, 1, 1, 10, 1, 1, 80, 80, 1, 1, 775, 1520, 775, 1, 1, 10915, 25945, 25945, 10915, 1, 1, 213720, 542910, 624670, 542910, 213720, 1, 1, 5245530, 14690640, 16408670, 16408670, 14690640, 5245530, 1, 1, 151534685, 479956020, 553630850, 464654480 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Row sums are:

{2, 2, 12, 162, 3072, 73722, 2137932, 72689682, 2834897592, 124735494042,

6112039208052,...}.

The sequence comes from a generalization of the recurrence for A008517.

The odd m;1,3,5,... share a fractal pattern modulo two:

a = Table[Table[e[n, k, m] + e[n, n - k, m], {k, 0, n}], {n, 0, 16}];

b = Table[If[m <= n, Mod[a[[n]][[m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];

ListDensityPlot[b, Mesh -> False, Frame -> False]

REFERENCES

Weisstein, Eric W. "Second-Order Eulerian Triangle." http://mathworld.wolfram.com/Second-OrderEulerianTriangle.html

LINKS

Table of n, a(n) for n=0..40.

FORMULA

m=5; e(n,k,n)=(k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m);

t(n,k)=e(n,k,m)+e(n,n-k,m).

EXAMPLE

{2},

{1, 1},

{1, 10, 1},

{1, 80, 80, 1},

{1, 775, 1520, 775, 1},

{1, 10915, 25945, 25945, 10915, 1},

{1, 213720, 542910, 624670, 542910, 213720, 1},

{1, 5245530, 14690640, 16408670, 16408670, 14690640, 5245530, 1},

{1, 151534685, 479956020, 553630850, 464654480, 553630850, 479956020, 151534685, 1},

{1, 4997226965, 17868940935, 22939735190, 16561843930, 16561843930, 22939735190, 17868940935, 4997226965, 1},

{1, 184877866870, 738031761670, 1076122652135, 791083062380, 531808521940, 791083062380, 1076122652135, 738031761670, 184877866870, 1}

MATHEMATICA

m = 5; e[n_, 0, m_] := 1;

e[n_, k_, m_] := 0 /; k >= n;

e[n_, k_, 1] := 1 /; k >= n;

e[n_, k_, m_] := (k + m - 1)e[n - 1, k, m] + (m*n - k + 1 - m)e[n - 1, k - 1, m];

Table[Table[e[n, k, m], {k, 0, n - 1}], {n, 1, 10}];

Table[Table[e[n, k, m] + e[n, n - k, m], {k, 0, n}], {n, 0, 10}];

Flatten[%]

CROSSREFS

A054091, A054090, A008517, A156141

Sequence in context: A288972 A065521 A225700 * A179930 A007737 A229243

Adjacent sequences:  A156185 A156186 A156187 * A156189 A156190 A156191

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Feb 05 2009

STATUS

approved

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Last modified April 11 15:49 EDT 2021. Contains 342886 sequences. (Running on oeis4.)