|
%I #16 Jul 30 2023 02:15:07
%S 1,1764,1942416,1744835904,1413470290176,1083688832185344,
%T 806595068762689536,590914962115587293184,429295503918929370218496,
%U 310518802877016005311463424,224098118280955193084850733056
%N Sixth column of triangle A112492 (inverse scaled Pochhammer symbols).
%C Also continuation of family of differences of reciprocals of unity. See A001242 and triangle A008969.
%H G. C. Greubel, <a href="/A111886/b111886.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..6} (1-6!*x/j).
%F a(n) = -((6!)^n)*Sum_{j=1..6} (-1)^j*binomial(6, j)/j^n, n >= 0.
%F a(n) = A112492(n+5, 6), 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+5,5], {n,0,30}] (* _G. C. Greubel_, Jul 24 2023 *)
%o (PARI) a(n) = -((6!)^n)*sum(j=1, 6, (-1)^j*binomial(6, j)/j^n); \\ _Michel Marcus_, Apr 28 2020
%o (Magma)
%o A111886:= func< n | (-1)*Factorial(6)^n*(&+[(-1)^j*Binomial(6,j)/j^n : j in [1..6]]) >;
%o [A111886(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 A111886(n): return T(n+5,5)
%o [A111886(n) for n in range(31)] # _G. C. Greubel_, Jul 24 2023
%Y Also right-hand column 5 in triangle A008969.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 12 2005
| 