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

1,2

Comments

A095267 has the same recursion rule but is derived from the matrix derived from A056939 (a type of generalized Narayana triangle).

Links

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

Cf. A095267, A001263.

Sequence in context: A233407 A272139 A167654 * A232338 A252937 A033277

Adjacent sequences: A095263 A095264 A095265 * A095267 A095268 A095269

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