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A095266
A sequence generated from the Narayana triangle considered as a matrix, or from Pascal's triangle.
1
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
OFFSET
1,2
COMMENTS
A095267 has the same recursion rule but is derived from the matrix derived from A056939 (a type of generalized Narayana triangle).
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 be 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.
From Colin Barker, Oct 21 2012: (Start)
a(n) = (n*(-8 + 25*n - 30*n^2 + 15*n^3))/2.
G.f.: -x*(39*x^3 + 103*x^2 + 37*x + 1)/(x-1)^5. (End)
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.
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}] (* Robert G. Wilson v, Jun 05 2004 *)
CROSSREFS
Sequence in context: A233407 A272139 A167654 * A232338 A252937 A033277
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, May 31 2004
EXTENSIONS
Edited and corrected by Robert G. Wilson v, Jun 05 2004
Typo in recurrence fixed by Colin Barker, Oct 21 2012
STATUS
approved