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A094952 A sequence derived from pentagonal numbers, or a Stirling number of the first kind matrix.

%I A094952 #19 Jun 01 2025 03:22:00
%S A094952 6,35,105,234,440,741,1155,1700,2394,3255,4301,5550,7020,8729,10695,
%T A094952 12936,15470,18315,21489,25010,28896,33165,37835,42924,48450,54431,
%U A094952 60885,67830,75284,83265,91791,100880,110550,120819,131705,143226,155400,168245,181779,196020
%N A094952 A sequence derived from pentagonal numbers, or a Stirling number of the first kind matrix.
%D A094952 Ruben Aldrovandi, Special Matrices of Mathematical Physics, World Scientific, 2001, 13.3.1 "Inverting Bell Matrices", p. 171.
%H A094952 Milan Janjic and Boris Petkovic, <a href="https://arxiv.org/abs/1301.4550">A Counting Function, arXiv:1301.4550 [math.CO], 2013.
%F A094952 a(n) = (2n+1)*A005449(n) where A005449 = 2, 7, 15, 26, 40, ...
%F A094952 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 A094952 Empirical g.f.: x*(6+11*x+x^2)/(1-x)^4. - _Colin Barker_, Jan 14 2012
%F A094952 From _Amiram Eldar_, Jun 01 2025: (Start)
%F A094952 Sum_{n>=1} 1/a(n) = 10 - sqrt(3)*Pi + 8*log(2) - 9*log(3).
%F A094952 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(3)-1)*Pi + 8*log(2) - 10. (End)
%e A094952 a(5) = 440 = (2n+1)*A005449(n) = 11 * 40.
%e A094952 a(6) = 741 since M^7 * [1 0 0 0] = [1 -6 57 -741].
%t A094952 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 *)
%t A094952 a[n_] := n*(2*n + 1)*(3*n + 1)/2; Array[a, 50] (* _Amiram Eldar_, Jun 01 2025 *)
%o A094952 (PARI) a(n) = n*(2*n + 1)*(3*n + 1)/2; \\ _Amiram Eldar_, Jun 01 2025
%Y A094952 Cf. A005449.
%K A094952 nonn
%O A094952 1,1
%A A094952 _Gary W. Adamson_, May 26 2004
%E A094952 Edited by _Robert G. Wilson v_, Jun 05 2004