A045756 Expansion of e.g.f. (1-9*x)^(-1/9), 9-factorial numbers.

1, 1, 10, 190, 5320, 196840, 9054640, 498005200, 31872332800, 2326680294400, 190787784140800, 17361688356812800, 1736168835681280000, 189242403089259520000, 22330603564532623360000, 2835986652695643166720000, 385694184766607470673920000, 55925656791158083247718400000

Offset

0,3

Comments

Nine-fold factorials of numbers 9k+1, k = 0, 1, 2, ... - M. F. Hasler, Feb 14 2020

Links

G. C. Greubel, Table of n, a(n) for n = 0..325 [a(0)=1 inserted by Georg Fischer, Feb 15 2020]

Peter Luschny, Mulitfactorials.

Index entries for sequences related to factorial numbers.

Formula

a(n+1) = (9*n+1)(!^9) = Product_{k=0..n-1} (9*k+1), n >= 0.

E.g.f. (1-9*x)^(-1/9).

D-finite with recurrence: a(n) +(-9*n+8)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

a(n) = A114806(9n-8). - M. F. Hasler, Feb 14 2020

a(n) = Sum_{k = 0..n} (-9)^(n - k) * A048994(n, k) = Sum_{k = 0..n} 9^(n - k) * A132393(n, k). Philippe Deléham, Sep 20 2008

a(n) = (-8)^n * sum_{k = 0..n} (9/8)^k * s(n + 1, n + 1 - k), where s(n, k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012

a(n) = 9^n * Gamma(n + 1/9) / Gamma(1/9). - Artur Jasinski Aug 23 2016

a(n) ~ sqrt(2 * Pi) * 9^n * n^(n - 7/18)/(Gamma(1/9) * exp(n)). - Ilya Gutkovskiy, Sep 10 2016

Sum_{n>=0} 1/a(n) = 1 + (e/9^8)^(1/9)*(Gamma(1/9) - Gamma(1/9, 1/9)). - Amiram Eldar, Dec 21 2022

Maple

seq( mul(9*j+1, j=0..n-1), n=0..20); # G. C. Greubel, Nov 11 2019

Mathematica

Table[9^n*Pochhammer[1/9, n], {n, 0, 20}] (* G. C. Greubel, Nov 11 2019 *)

Prog

(PARI) vector(21, n, prod(j=0, n-2, 9*j+1) ) \\ G. C. Greubel, Nov 11 2019
(Magma) [1] cat [(&*[9*j+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019
(Sage) [product( (9*j+1) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 11 2019
(GAP) List([0..20], n-> Product([0..n-1], j-> 9*j+1) ); # G. C. Greubel, Nov 11 2019

Crossrefs

Cf. A008542, A048994, A114806 (9-fold factorials), A132393.

Cf. k-fold factorials : A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A144773 (10), A256268 (combined table).

Sequence in context: A113373 A211826 A144772 * A072387 A356495 A244385

Adjacent sequences: A045753 A045754 A045755 * A045757 A045758 A045759

Keyword

easy,nonn

Author

Wolfdieter Lang

Extensions

a(0)=1 inserted; merged with A144772; formulas and programs changed accordingly by Georg Fischer, Feb 15 2020

Status

approved