|
%I #41 Dec 21 2022 04:46:04
%S 1,1,10,190,5320,196840,9054640,498005200,31872332800,2326680294400,
%T 190787784140800,17361688356812800,1736168835681280000,
%U 189242403089259520000,22330603564532623360000,2835986652695643166720000,385694184766607470673920000,55925656791158083247718400000
%N Expansion of e.g.f. (1-9*x)^(-1/9), 9-factorial numbers.
%C Nine-fold factorials of numbers 9k+1, k = 0, 1, 2, ... - _M. F. Hasler_, Feb 14 2020
%H G. C. Greubel, <a href="/A045756/b045756.txt">Table of n, a(n) for n = 0..325</a> [a(0)=1 inserted by _Georg Fischer_, Feb 15 2020]
%H Peter Luschny, <a href="/wiki/User:Peter_Luschny/Multifactorials">Mulitfactorials</a>.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F a(n+1) = (9*n+1)(!^9) = Product_{k=0..n-1} (9*k+1), n >= 0.
%F E.g.f. (1-9*x)^(-1/9).
%F D-finite with recurrence: a(n) +(-9*n+8)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%F a(n) = A114806(9n-8). - _M. F. Hasler_, Feb 14 2020
%F 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
%F 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
%F a(n) = 9^n * Gamma(n + 1/9) / Gamma(1/9). - _Artur Jasinski_ Aug 23 2016
%F a(n) ~ sqrt(2 * Pi) * 9^n * n^(n - 7/18)/(Gamma(1/9) * exp(n)). - _Ilya Gutkovskiy_, Sep 10 2016
%F 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
%p seq( mul(9*j+1, j=0..n-1), n=0..20); # _G. C. Greubel_, Nov 11 2019
%t Table[9^n*Pochhammer[1/9, n], {n,0,20}] (* _G. C. Greubel_, Nov 11 2019 *)
%o (PARI) vector(21, n, prod(j=0,n-2, 9*j+1) ) \\ _G. C. Greubel_, Nov 11 2019
%o (Magma) [1] cat [(&*[9*j+1: j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Nov 11 2019
%o (Sage) [product( (9*j+1) for j in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Nov 11 2019
%o (GAP) List([0..20], n-> Product([0..n-1], j-> 9*j+1) ); # _G. C. Greubel_, Nov 11 2019
%Y Cf. A008542, A048994, A114806 (9-fold factorials), A132393.
%Y 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).
%K easy,nonn
%O 0,3
%A _Wolfdieter Lang_
%E a(0)=1 inserted; merged with A144772; formulas and programs changed accordingly by _Georg Fischer_, Feb 15 2020
|
