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A178347 A symmetrical triangle between the Eulerian numbers, A008292, and MacMahon numbers, A060187, by linear combination with Pascal(A007318), Eulerian numbers ,and Narayana numbers,A001263,:k=6;t(n,m,k)=Binomial[n, m] - k*(Binomial[n, m]*Binomial[n + 1, m]/(m + 1)) + k*Eulerian[n + 1, m] 0
1, 1, 1, 1, 8, 1, 1, 33, 33, 1, 1, 100, 282, 100, 1, 1, 257, 1522, 1522, 257, 1, 1, 600, 6531, 13466, 6531, 600, 1, 1, 1321, 24603, 90809, 90809, 24603, 1321, 1, 1, 2804, 85660, 522404, 926626, 522404, 85660, 2804, 1, 1, 5817, 283836, 2716116, 7830498 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 10, 68, 484, 3560, 27730, 233468, 2148364, 21672536, 239149108,...}.

By solving the linear combination of Pascal,Narayana and Eulerian to give

the MacMahon at k=4, I got this functional set of symmetrical triangles.

By modulo two pattern the k=3 set is Narayana numbers like and the k=6 ,{1.8.1} level,

is a new even Sierpinski type ( compare A142458).

LINKS

Table of n, a(n) for n=0..49.

FORMULA

k=6;

t(n,m,k)=Binomial[n, m] - k*(Binomial[n, m]*Binomial[n + 1, m]/(m + 1)) + k*Eulerian[n + 1, m]

EXAMPLE

{1},

{1, 1},

{1, 8, 1},

{1, 33, 33, 1},

{1, 100, 282, 100, 1},

{1, 257, 1522, 1522, 257, 1},

{1, 600, 6531, 13466, 6531, 600, 1},

{1, 1321, 24603, 90809, 90809, 24603, 1321, 1},

{1, 2804, 85660, 522404, 926626, 522404, 85660, 2804, 1},

{1, 5817, 283836, 2716116, 7830498, 7830498, 2716116, 283836, 5817, 1},

{1, 11896, 910917, 13191348, 58345734, 94229316, 58345734, 13191348, 910917, 11896, 1}

MATHEMATICA

<< DiscreteMath`Combinatorica`

t[n_, m_, k_] = Binomial[n, m] - k*(Binomial[n, m]* Binomial[n + 1, m]/(m + 1)) + k*Eulerian[n + 1, m];

Table[Flatten[Table[Table[t[n, m, k], {m, 0, n}], {n, 0, 10}]], {k, 0, 10}]

CROSSREFS

Cf. A007318, A008292, A001263, A060187, A142458

Sequence in context: A168523 A144439 A157208 * A141686 A185412 A157148

Adjacent sequences:  A178344 A178345 A178346 * A178348 A178349 A178350

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, May 25 2010

STATUS

approved

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Last modified November 14 10:04 EST 2019. Contains 329111 sequences. (Running on oeis4.)