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%I #15 Feb 02 2021 21:11:52
%S 1,42,303,1144,3105,6906,13447,23808,39249,61210,91311,131352,183313,
%T 249354,331815,433216,556257,703818,878959,1084920,1325121,1603162,
%U 1922823,2288064,2703025,3172026,3699567,4290328,4949169,5681130
%N A sequence generated from the Narayana triangle considered as a matrix, or from Pascal's triangle.
%C A095267 has the same recursion rule but is derived from the matrix derived from A056939 (a type of generalized Narayana triangle).
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F 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.
%F From _Colin Barker_, Oct 21 2012: (Start)
%F a(n) = (n*(-8 + 25*n - 30*n^2 + 15*n^3))/2.
%F G.f.: -x*(39*x^3 + 103*x^2 + 37*x + 1)/(x-1)^5. (End)
%e 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.
%t 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 *)
%Y Cf. A095267, A001263.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, May 31 2004
%E Edited and corrected by _Robert G. Wilson v_, Jun 05 2004
%E Typo in recurrence fixed by _Colin Barker_, Oct 21 2012
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