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A154353 Triangle T(n,m) read by rows: T(n,m) = ( Eulerian(n,m) - Binomial(n,m)^2 )/2, n >= 4, 2 <= m = <= n-1. 1
1, 1, 5, 15, 5, 16, 101, 101, 16, 42, 483, 1008, 483, 42, 99, 1926, 7197, 7197, 1926, 99, 219, 6912, 42549, 75645, 42549, 6912, 219, 466, 23272, 224068, 647239, 647239, 224068, 23272, 466, 968, 75306, 1094544, 4847007, 7830372, 4847007, 1094544, 75306, 968 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,3

COMMENTS

Row sums are: {2, 25, 234, 2058, 18444, 175005, 1790090, 19866022, 239148084, 3112158322, 43583945300,...}.

Noticing the Eulerian numbers and the binomial squared were the same for the first four rows, I subtracted them and extracted the zeros to get this sequence.

The resulting fractal can be obtained from the Mathematica code given in the Mathematica code section.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows

FORMULA

T(n,m) = ( Eulerian(n,m) - Binomial(n,m)^2 )/2 for n >= 4, and 2 <= m <= n-1.

EXAMPLE

{1, 1},

{5, 15, 5},

{16, 101, 101, 16},

{42, 483, 1008, 483, 42},

{99, 1926, 7197, 7197, 1926, 99},

{219, 6912, 42549, 75645, 42549, 6912, 219},

{466, 23272, 224068, 647239, 647239, 224068, 23272, 466},

{968, 75306, 1094544, 4847007, 7830372, 4847007, 1094544, 75306, 968},

{1981, 237623, 5080230, 33104787, 81149421, 81149421, 33104787, 5080230, 237623, 1981},

{4017, 737685, 22742525, 211518255, 752497122, 1137159114, 752497122, 211518255, 22742525, 737685, 4017},

{8100, 2265615, 99164495, 1285615475, 6420803247, 13984115718, 13984115718, 6420803247, 1285615475, 99164495, 2265615, 8100}

MATHEMATICA

p[x_, n_] = (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x; Table[Table[(CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m]] - Binomial[n - 1, m - 1]^2)/2, {m, 2, n - 1}], {n, 4, 14}]; Flatten[%]

(* fractal code *)

a = Table[Table[(CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m]] - Binomial[n - 1, m - 1]^2)/2, {m, 2, n - 1}], {n, 4, 34}];

b = Table[If[m <= n + 1, Mod[a[[n]][[m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}]; ListDensityPlot[b, Mesh -> False]

CROSSREFS

Sequence in context: A107776 A290528 A161202 * A114332 A077348 A290829

Adjacent sequences:  A154350 A154351 A154352 * A154354 A154355 A154356

KEYWORD

nonn,tabf

AUTHOR

Roger L. Bagula, Jan 07 2009

STATUS

approved

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)