A095266 A sequence generated from the Narayana triangle considered as a matrix, or from Pascal's triangle.
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
Examples
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.
Links
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Programs
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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}] (* Robert G. Wilson v, Jun 05 2004 *)
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)
Extensions
Edited and corrected by Robert G. Wilson v, Jun 05 2004
Typo in recurrence fixed by Colin Barker, Oct 21 2012
Comments