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Eighth column of triangle A112492 (inverse scaled Pochhammer symbols).
2

%I #15 Jul 25 2023 09:15:17

%S 1,109584,7245893376,381495483224064,17810567950611972096,

%T 778101042571221893382144,32762625292956765972873609216,

%U 1351813956241264848815287984717824

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

%C Also continuation of family of Differences of reciprocals of unity. See A001242, A111887 and triangle A008969.

%H 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 G.f.: 1/Product_{j=1..8} 1-8!*x/j.

%F a(n) = -((8!)^n) * Sum_{j=1..8} (-1)^j*binomial(8, j)/j^n, n>=0.

%F a(n) = A112492(n+7, 8), n>=0.

%t 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 Table[T[n+7,7], {n,0,30}] (* _G. C. Greubel_, Jul 24 2023 *)

%o (PARI) a(n) = -((8!)^n)*sum(j=1, 8, ((-1)^j)*binomial(8, j)/j^n); \\ _Michel Marcus_, Apr 28 2020

%o (Magma)

%o A111888:= func< n | (-1)*Factorial(8)^n*(&+[(-1)^j*Binomial(8,j)/j^n : j in [1..8]]) >;

%o [A111888(n): n in [0..30]]; // _G. C. Greubel_, Jul 24 2023

%o (SageMath)

%o @CachedFunction

%o def T(n,k): # T = A112492

%o if (k==0 or k==n): return 1

%o else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)

%o def A111888(n): return T(n+7,7)

%o [A111888(n) for n in range(31)] # _G. C. Greubel_, Jul 24 2023

%Y Also right-hand column 7 in triangle A008969.

%Y Cf. A001242, A111887, A112492.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Sep 12 2005