cp's OEIS Frontend

A111888 Eighth column of triangle A112492 (inverse scaled Pochhammer symbols).

%I A111888 #15 Jul 25 2023 09:15:17
%S A111888 1,109584,7245893376,381495483224064,17810567950611972096,
%T A111888 778101042571221893382144,32762625292956765972873609216,
%U A111888 1351813956241264848815287984717824
%N A111888 Eighth column of triangle A112492 (inverse scaled Pochhammer symbols).
%C A111888 Also continuation of family of Differences of reciprocals of unity. See A001242, A111887 and triangle A008969.
%H A111888 Mircea Merca, <a href="https://www.researchgate.net/publication/264664262_Some_experiments_with_complete_and_elementary_symmetric_functions">Some experiments with complete and elementary symmetric functions</a>, Periodica Mathematica Hungarica, 69 (2014), 182-189.
%F A111888 G.f.: 1/Product_{j=1..8} 1-8!*x/j.
%F A111888 a(n) = -((8!)^n) * Sum_{j=1..8} (-1)^j*binomial(8, j)/j^n, n>=0.
%F A111888 a(n) = A112492(n+7, 8), n>=0.
%t A111888 T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, (k+1)^(n-k)*T[n-1,k-1] +k!*T[n-1,k]]; (* T = A112492 *)
%t A111888 Table[T[n+7,7], {n,0,30}] (* _G. C. Greubel_, Jul 24 2023 *)
%o A111888 (PARI) a(n) = -((8!)^n)*sum(j=1, 8, ((-1)^j)*binomial(8, j)/j^n); \\ _Michel Marcus_, Apr 28 2020
%o A111888 (Magma)
%o A111888 A111888:= func< n | (-1)*Factorial(8)^n*(&+[(-1)^j*Binomial(8,j)/j^n : j in [1..8]]) >;
%o A111888 [A111888(n): n in [0..30]]; // _G. C. Greubel_, Jul 24 2023
%o A111888 (SageMath)
%o A111888 @CachedFunction
%o A111888 def T(n,k): # T = A112492
%o A111888     if (k==0 or k==n): return 1
%o A111888     else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
%o A111888 def A111888(n): return T(n+7,7)
%o A111888 [A111888(n) for n in range(31)] # _G. C. Greubel_, Jul 24 2023
%Y A111888 Also right-hand column 7 in triangle A008969.
%Y A111888 Cf. A001242, A111887, A112492.
%K A111888 nonn,easy
%O A111888 0,2
%A A111888 _Wolfdieter Lang_, Sep 12 2005