%I #33 Dec 22 2022 04:15:28

%S 1,10,210,6720,288960,15603840,1014249600,77082969600,6706218355200,

%T 657209398809600,71635824470246400,8596298936429568000,

%U 1126115160672273408000,159908352815462823936000,24465977980765812062208000,4012420388845593178202112000

%N Expansion of e.g.f.: (1-11*x)^(-10/11).

%C Generally, for k > 1, if e.g.f. = (1-k*x)^(-(k-1)/k) then a(n) ~ n! * k^n / (n^(1/k) * Gamma((k-1)/k)).

%H G. C. Greubel, <a href="/A254322/b254322.txt">Table of n, a(n) for n = 0..300</a>

%F D-finite with recurrence: a(0) = 1; a(n) = (11*n-1) * a(n-1) for n > 0. [corrected by _Georg Fischer_, Dec 23 2019]

%F a(n) = 11^n * Gamma(n+10/11) / Gamma(10/11).

%F a(n) ~ n! * 11^n / (n^(1/11) * Gamma(10/11)).

%F From _Nikolaos Pantelidis_, Jan 17 2021: (Start)

%F G.f.: 1/G(0) where G(k) = 1 - (22*k+10)*x - 11*(k+1)*(11*k+10)*x^2/G(k+1) (continued fraction).

%F G.f.: 1/(1-10*x-110*x^2/(1-32*x-462*x^2/(1-54*x-1056*x^2/(1-76*x-1892*x^2/(1-98*x-2970*x^2/(1-...)))))) (Jacobi continued fraction).

%F G.f.: 1/Q(0) where Q(k) = 1 - x*(11*k+10)/(1 - x*(11*k+11)/Q(k+1)) (continued fraction).

%F G.f.: 1/(1-10*x/(1-11*x/(1-21*x/(1-22*x/(1-32*x/(1-33*x/(1-43*x/(1-44*x/(1-54*x/(1-55*x/(1-...))))))))))) (Stieltjes continued fraction).

%F (End)

%F G.f.: hypergeometric2F0([1, 10/11], [--], 11*x). - _G. C. Greubel_, Feb 08 2022

%F Sum_{n>=0} 1/a(n) = 1 + (e/11)^(1/11)*(Gamma(10/11) - Gamma(10/11, 1/11)). - _Amiram Eldar_, Dec 22 2022

%t CoefficientList[Series[(1-11*x)^(-10/11), {x, 0, 20}], x] * Range[0, 20]!

%t FullSimplify[Table[11^n * Gamma[n+10/11] / Gamma[10/11], {n, 0, 18}]]

%o (Magma) m=11; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // _G. C. Greubel_, Feb 08 2022

%o (Sage) m=11; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # _G. C. Greubel_, Feb 08 2022

%Y Sequences of the form k^n*Pochhammer((k-1)/k, n): A000007 (k=1), A001147 (k=2), A008544 (k=3), A008545 (k=4), A008546 (k=5), A008543 (k=6), A049209 (k=7), A049210 (k=8), A049211 (k=9), A049212 (k=10), this sequence (k=11), A346896 (k=12).

%Y Cf. A000108, A254282, A254286, A254287.

%K nonn,easy

%O 0,2

%A _Vaclav Kotesovec_, Jan 28 2015