A111888 Eighth column of triangle A112492 (inverse scaled Pochhammer symbols).
1, 109584, 7245893376, 381495483224064, 17810567950611972096, 778101042571221893382144, 32762625292956765972873609216, 1351813956241264848815287984717824
Offset: 0
Links
- Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.
Programs
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Magma
A111888:= func< n | (-1)*Factorial(8)^n*(&+[(-1)^j*Binomial(8,j)/j^n : j in [1..8]]) >; [A111888(n): n in [0..30]]; // G. C. Greubel, Jul 24 2023
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Mathematica
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 *) Table[T[n+7,7], {n,0,30}] (* G. C. Greubel, Jul 24 2023 *)
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PARI
a(n) = -((8!)^n)*sum(j=1, 8, ((-1)^j)*binomial(8, j)/j^n); \\ Michel Marcus, Apr 28 2020
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SageMath
@CachedFunction def T(n,k): # T = A112492 if (k==0 or k==n): return 1 else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k) def A111888(n): return T(n+7,7) [A111888(n) for n in range(31)] # G. C. Greubel, Jul 24 2023
Formula
G.f.: 1/Product_{j=1..8} 1-8!*x/j.
a(n) = -((8!)^n) * Sum_{j=1..8} (-1)^j*binomial(8, j)/j^n, n>=0.
a(n) = A112492(n+7, 8), n>=0.
Comments