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%I #18 Jul 25 2023 09:15:13
%S 1,13068,104587344,673781602752,3878864920694016,21006340945438768128,
%T 110019668725577574273024,565858042127972959667208192,
%U 2882220940619488483325345857536,14605752814655604919042956624396288
%N Seventh column of triangle A112492 (inverse scaled Pochhammer symbols).
%C Also continuation of family of differences of reciprocals of unity. See A001242, A111886 and triangle A008969.
%H G. C. Greubel, <a href="/A111887/b111887.txt">Table of n, a(n) for n = 0..250</a>
%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..7} 1-7!*x/j.
%F a(n) = -((7!)^n) * Sum_{j=1..7} (-1)^j*binomial(7, j)/j^n, n>=0.
%F a(n) = A112492(n+6, 7), 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+6,6], {n,0,30}] (* _G. C. Greubel_, Jul 24 2023 *)
%o (PARI) a(n) = -((7!)^n)*sum(j=1, 7, ((-1)^j)*binomial(7, j)/j^n); \\ _Michel Marcus_, Apr 28 2020
%o (Magma)
%o A111887:= func< n | (-1)*Factorial(7)^n*(&+[(-1)^j*Binomial(7,j)/j^n : j in [1..7]]) >;
%o [A111887(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 A111887(n): return T(n+6,6)
%o [A111887(n) for n in range(31)] # _G. C. Greubel_, Jul 24 2023
%Y Also right-hand column 6 in triangle A008969.
%Y Cf. A001242, A111886, A112492.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 12 2005
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