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A124790
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A generalized Motzkin triangle.
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2
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1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 3, 4, 3, 2, 1, 0, 6, 9, 6, 5, 2, 1, 0, 15, 21, 15, 12, 6, 3, 1, 0, 36, 51, 36, 30, 15, 9, 3, 1, 0, 91, 127, 91, 76, 40, 25, 10, 4, 1, 0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1
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OFFSET
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0,13
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COMMENTS
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Columns include A005043, A001006, A002026. Row sums are A124791. For even k, column k has g.f. x^k*M(x)^(k/2), where M(x)=2/(1-x+sqrt(1-2x-3x^2)) is the g.f. of A001006. For odd k, column k has g.f. x^k*S(x)*M(x)^floor(k/2), S(x)=(1+x-sqrt(1-2x-3x^2))/(2x(1+x)), the g.f. of A005043.
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LINKS
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E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
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FORMULA
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Triangle is the product of A124788 and A124305, that is, it is the product of the expansion of (1+x*y)/(1-x^2*y^2-x^3*y^2) and the inverse of the Riordan array (1,x(1-x^2)).
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EXAMPLE
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Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 1, 2, 1, 1,
0, 3, 4, 3, 2, 1,
0, 6, 9, 6, 5, 2, 1,
0, 15, 21, 15, 12, 6, 3, 1,
0, 36, 51, 36, 30, 15, 9, 3, 1,
0, 91, 127, 91, 76, 40, 25, 10, 4, 1,
0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1
Production matrix begins
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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