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A089901 Main diagonal of A089900, also the inverse hyperbinomial of A000312 (offset 1).

%I A089901 #30 Oct 23 2024 16:06:14
%S A089901 1,3,18,159,1848,26595,456048,9073911,205437312,5214027267,
%T A089901 146602156800,4522866752943,151895344131072,5516066815430691,
%U A089901 215373243256915968,8996883483108522375,400372897193449586688,18908951043963993686019
%N A089901 Main diagonal of A089900, also the inverse hyperbinomial of A000312 (offset 1).
%C A089901 The n-th row of array A089900 is the n-th binomial transform of the factorials found in row 0: {1!,2!,3!,...,(n+1)!,...}. The hyperbinomial transform of this main diagonal gives: {1,4,27,...,(n+1)^(n+1),...}, which is the next lower diagonal in array A089900.
%C A089901 a(n), for n>=1, is the number of colored labeled mappings from n points to themselves, where each component is one of three colors. - _Steven Finch_, Nov 28 2021
%H A089901 G. C. Greubel, <a href="/A089901/b089901.txt">Table of n, a(n) for n = 0..380</a>
%H A089901 Nicholas John Bizzell-Browning, <a href="https://bura.brunel.ac.uk/handle/2438/29960">LIE scales: Composing with scales of linear intervallic expansion</a>, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 162.
%H A089901 Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.
%F A089901 a(n) = Sum_{k=0..n} n^(n-k) * C(n,k) * (k+1)!.
%F A089901 a(n) = Sum_{k=0..n} -(n-k-1)^(n-k-1) * C(n,k) * (k+1)^(k+1).
%F A089901 E.g.f.: 1 / (1 + LambertW(-x))^3.
%F A089901 E.g.f.: (Sum_{n>=0} (n+1)^(n+1) * x^n/n!) * (Sum_{n>=0} -(n-1)^(n-1) * x^n/n!).
%F A089901 a(n) ~ n^(n+1) * (1 + sqrt(Pi/(2*n))). - _Vaclav Kotesovec_, Jul 09 2013
%F A089901 a(n) = (n^(n + 2) + exp(n)*Gamma(n + 2, n)) / (n + 1). - _Peter Luschny_, Nov 29 2021
%t A089901 CoefficientList[Series[1/(1+LambertW[-x])^3, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jul 09 2013 *)
%t A089901 Flatten[{1,Table[Sum[n^(n-k)*Binomial[n,k]*(k+1)!,{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Jul 09 2013 *)
%t A089901 a[n_] := (n^(n + 2) + Exp[n] Gamma[n + 2, n]) / (n + 1);
%t A089901 Table[a[n], {n, 0, 17}]  (* _Peter Luschny_, Nov 29 2021 *)
%o A089901 (PARI) /* As (n+1)-th term of the n-th binomial transform of {(n+1)!}: */
%o A089901 {a(n)=if(n<0,0,sum(i=0,n,n^(n-i)*binomial(n,i)*(i+1)!))}
%o A089901 for(n=0,20,print1(a(n),", "))
%o A089901 (PARI) /* As (n+1)-th term of inverse hyperbinomial of {(n+1)^(n+1)}: */
%o A089901 {a(n)=if(n<0,0,sum(i=0,n,-(n-i-1)^(n-i-1)*binomial(n,i)*(i+1)^(i+1)))}
%o A089901 for(n=0,20,print1(a(n),", "))
%Y A089901 Cf. A000312, A089900, A089902, A000312.
%K A089901 nonn
%O A089901 0,2
%A A089901 _Paul D. Hanna_, Nov 14 2003