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A095266
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A sequence generated from the Narayana triangle considered as a matrix, or from Pascal's triangle.
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1
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1, 42, 303, 1144, 3105, 6906, 13447, 23808, 39249, 61210, 91311, 131352, 183313, 249354, 331815, 433216, 556257, 703818, 878959, 1084920, 1325121, 1603162, 1922823, 2288064, 2703025, 3172026, 3699567, 4290328, 4949169, 5681130
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A095267 has the same recursion rule but is derived from the matrix derived from A056939 (a type of generalized Narayana triangle).
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FORMULA
| a(n+6) = 5*a(n+5) - 10*a(n+4) + 10*a(n+3) - 5*a(n+2) - a(n); where the multipliers with changed signs are found in the characteristic polynomial of the generating matrix M: x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1. Let M = the 5th order Matrix M, having Narayana triangle (A001263) rows (fill in with zeros): [1 0 0 0 0 / 1 1 0 0 0 / 1 3 1 0 0 / 1 6 6 1 0 / 1 10 20 10 1]. Then M^n *[1 0 0 0 0] = [1 n A000326(n) A005915(n) a(n)] where A000326 = the pentagonal numbers and A005915 = the hex prism numbers.
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EXAMPLE
| a(7) = 23808 = 5*a(6) - 10*a(5) + 10*a(4) - 5*a(3) + a(2) = 5*13447 - 10*6906 + 10*3105 - 5*1144 + 303.
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MATHEMATICA
| a[n_] := (MatrixPower[{{1, 0, 0, 0, 0}, {1, 1, 0, 0, 0}, {1, 3, 1, 0, 0}, {1, 6, 6, 1, 0}, {1, 10, 20, 10, 1}}, n].{{1}, {0}, {0}, {0}, {0}})[[5, 1]]; Table[ a[n], {n, 30}] (from Robert G. Wilson v Jun 05 2004)
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CROSSREFS
| Cf. A095267, A001263.
Sequence in context: A036463 A190534 A167654 * A033277 A190533 A190602
Adjacent sequences: A095263 A095264 A095265 * A095267 A095268 A095269
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), May 31 2004
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EXTENSIONS
| Edited and corrected by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 05 2004
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