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%I #15 Aug 12 2023 21:10:39
%S 6,35,105,234,440,741,1155,1700,2394,3255,4301,5550,7020,8729,10695,
%T 12936,15470,18315,21489,25010,28896,33165,37835,42924,48450,54431,
%U 60885,67830,75284,83265,91791,100880,110550,120819,131705,143226
%N A sequence derived from pentagonal numbers, or a Stirling number of the first kind matrix.
%D R. Aldrovandi, "Special Matrices of Mathematical Physics", World Scientific, 2001, 13.3.1 "Inverting Bell Matrices", p. 171.
%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550, 2013
%F a(n) = (2n+1)*A005449(n) where A005449 = 2, 7, 15, 26, 40, ...
%F Given the 4th-order Stirling number of the first kind matrix [1 0 0 0 / -1 1 0 0 / 2 -3 1 0 / -6 11 -6 1] = M, M^n * [1 0 0 0] = [1 -n A005449(n) -a(n)].
%F Empirical g.f.: x*(6+11*x+x^2)/(1-x)^4. - _Colin Barker_, Jan 14 2012
%e a(5) = 440 = (2n+1)*A005449(n) = 11 * 40.
%e a(6) = 741 since M^7 * [1 0 0 0] = [1 -6 57 -741].
%t a[n_] := (MatrixPower[{{1, 0, 0, 0}, {-1, 1, 0, 0}, {2, -3, 1, 0}, {-6, 11, -6, 1}}, n].{{1}, {0}, {0}, {0}})[[4, 1]]; Table[ Abs[ a[n]], {n, 36}] (* _Robert G. Wilson v_, Jun 05 2004 *)
%Y Cf. A005449.
%K nonn
%O 1,1
%A _Gary W. Adamson_, May 26 2004
%E Edited by _Robert G. Wilson v_, Jun 05 2004
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