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

6, 35, 105, 234, 440, 741, 1155, 1700, 2394, 3255, 4301, 5550, 7020, 8729, 10695, 12936, 15470, 18315, 21489, 25010, 28896, 33165, 37835, 42924, 48450, 54431, 60885, 67830, 75284, 83265, 91791, 100880, 110550, 120819, 131705, 143226, 155400, 168245, 181779, 196020

Offset: 1

Author

Gary W. Adamson, May 26 2004

Keywords

• nonn

Examples

			a(5) = 440 = (2n+1)*A005449(n) = 11 * 40.
a(6) = 741 since M^7 * [1 0 0 0] = [1 -6 57 -741].
		

References

• Ruben Aldrovandi, Special Matrices of Mathematical Physics, World Scientific, 2001, 13.3.1 "Inverting Bell Matrices", p. 171.

Links

• Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.

Crossrefs

Cf. A005449.

Programs

Mathematica

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 *)
a[n_] := n*(2*n + 1)*(3*n + 1)/2; Array[a, 50] (* Amiram Eldar, Jun 01 2025 *)

PARI

a(n) = n*(2*n + 1)*(3*n + 1)/2; \\ Amiram Eldar, Jun 01 2025

Formula

a(n) = (2n+1)*A005449(n) where A005449 = 2, 7, 15, 26, 40, ...

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)].

Empirical g.f.: x*(6+11*x+x^2)/(1-x)^4. - Colin Barker, Jan 14 2012

From Amiram Eldar, Jun 01 2025: (Start)

Sum_{n>=1} 1/a(n) = 10 - sqrt(3)*Pi + 8*log(2) - 9*log(3).

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(3)-1)*Pi + 8*log(2) - 10. (End)

Extensions

Edited by Robert G. Wilson v, Jun 05 2004