A035097 Related to 9-factorial numbers A045756.

1, 45, 2565, 161595, 10762227, 742593663, 52511980455, 3780862592760, 276002969271480, 20369019132235224, 1516566060845513496, 113742454563413512200, 8583180609746819651400, 651095557682223033556200, 49613481495385395156982440, 3795431334396982729509156660

Offset

1,2

Comments

Convolution of A035024(n-1) with A025754(n), n >= 1.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..526

Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Index entries for sequences related to factorial numbers.

Formula

a(n) = 9^(n-1)*A045756(n)/n!, where A045756(n) = (9*n-8)(!^9) = Product_{j=1..n} (9*j-8).

G.f.: (-1 + (1-81*x)^(-1/9))/9.

D-finite with recurrence: n*a(n) - 9*(9*n-8)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020

a(n) ~ 9^(2*n-1) * n^(-8/9) / Gamma(1/9). - Amiram Eldar, Aug 18 2025

Mathematica

CoefficientList[Series[(1/(1-81 x)^(1/9)-1)/(9 x), {x, 0, 20}], x] (* Harvey P. Dale, May 14 2011 *)
Table[9^(2*n-1)*Pochhammer[1/9, n]/n!, {n, 20}] (* G. C. Greubel, Dec 30 2025 *)

Prog

(Magma)
A035097:= func< n | 9^(n-1)*(&*[9*j-8: j in [1..n]])/Factorial(n) >;
[A035097(n): n in [1..30]]; // G. C. Greubel, Dec 30 2025
(SageMath)
def A035097(n): return 9^(2*n-1)*rising_factorial(1/9, n)/factorial(n)
print([A035097(n) for n in range(1, 31)]) # G. C. Greubel, Dec 30 2025

Crossrefs

Cf. A025754, A034996, A035024, A045756, A256190.

Sequence in context: A279439 A251443 A109941 * A184286 A130017 A025755

Adjacent sequences: A035094 A035095 A035096 * A035098 A035099 A035100

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved