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

1,1

References

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

Links

Table of n, a(n) for n=1..40.

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

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)

Example

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

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 *)

Prog

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

Crossrefs

Cf. A005449.

Sequence in context: A370091 A033578 A101077 * A024526 A213504 A089581

Adjacent sequences: A094949 A094950 A094951 * A094953 A094954 A094955

Keyword

nonn

Author

Gary W. Adamson, May 26 2004

Extensions

Edited by Robert G. Wilson v, Jun 05 2004

Status

approved