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 A178346 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=3;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, 5, 1, 1, 18, 18, 1, 1, 52, 144, 52, 1, 1, 131, 766, 766, 131, 1, 1, 303, 3273, 6743, 3273, 303, 1, 1, 664, 12312, 45422, 45422, 12312, 664, 1, 1, 1406, 42844, 261230, 463348, 261230, 42844, 1406, 1, 1, 2913, 141936, 1358100, 3915312 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {1, 2, 7, 38, 250, 1796, 13897, 116798, 1074310, 10836524, 119575066,...}. 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. LINKS FORMULA k=3; 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, 5, 1}, {1, 18, 18, 1}, {1, 52, 144, 52, 1}, {1, 131, 766, 766, 131, 1}, {1, 303, 3273, 6743, 3273, 303, 1}, {1, 664, 12312, 45422, 45422, 12312, 664, 1}, {1, 1406, 42844, 261230, 463348, 261230, 42844, 1406, 1}, {1, 2913, 141936, 1358100, 3915312, 3915312, 1358100, 141936, 2913, 1}, {1, 5953, 455481, 6595734, 29172972, 47114784, 29172972, 6595734, 455481, 5953, 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 Sequence in context: A157181 A029847 A154334 * A168551 A262307 A144397 Adjacent sequences:  A178343 A178344 A178345 * A178347 A178348 A178349 KEYWORD nonn,tabl,uned AUTHOR Roger L. Bagula, May 25 2010 STATUS approved

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Last modified October 20 12:47 EDT 2019. Contains 328257 sequences. (Running on oeis4.)