A346896 - OEIS

%I #23 Dec 22 2022 04:15:24

%S 1,11,253,8855,416185,24554915,1743398965,144702114095,13746700839025,

%T 1470896989775675,175036741783305325,22929813173612997575,

%U 3278963283826658653225,508239308993132091249875,84875964601853059238729125,15192797663731697603732513375

%N Expansion of e.g.f.: (1-12*x)^(-11/12).

%H G. C. Greubel, <a href="/A346896/b346896.txt">Table of n, a(n) for n = 0..300</a>

%F G.f.: 1/(1-11*x/(1-12*x/(1-23*x/(1-24*x/(1-35*x/(1-36*x/(1-47*x/(1-48*x/(1-59*x/(1-60*x/(1-...))))))))))) (Stieltjes continued fraction).

%F G.f.: 1/Q(0) where Q(k) = 1 - x*(12*k+11)/(1 - x*(12*k+12)/Q(k+1) ) (continued fraction).

%F G.f.: 1/(1-11*x-132*x^2/(1-35*x-552*x^2/(1-59*x-1260*x^2/(1-83*x-2256*x^2/(1-107*x-3540*x^2/(1-...)))))) (Jacobi continued fraction).

%F G.f.: 1/G(0) where G(k) = 1 - x*(24*k+11) - 12*(k+1)*(12*k+11)*x^2/G(k+1) (continued fraction).

%F a(n) = 12^n*Gamma(n+11/12)/Gamma(11/12). - _Stefano Spezia_, Aug 07 2021

%F Sum_{n>=0} 1/a(n) = 1 + (e/12)^(1/12)*(Gamma(11/12) - Gamma(11/12, 1/12)). - _Amiram Eldar_, Dec 22 2022

%t CoefficientList[Series[(1-12*x)^(-11/12),{x,0,20}], x] * Range[0, 20]!

%t FullSimplify[Table[12^n Gamma[n+11/12]/Gamma[11/12],{n,0,15}]] (* _Stefano Spezia_, Aug 07 2021 *)

%o (Sage) m=12; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # _G. C. Greubel_, Feb 16 2022

%o (Magma) m:=12; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // _G. C. Greubel_, Feb 16 2022

%Y Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), this sequence (m=12).

%K nonn,easy

%O 0,2

%A _Nikolaos Pantelidis_, Aug 06 2021