|
%I #16 Jun 27 2022 19:03:49
%S 0,3,13,292,5511,166091,6096546,281962395,15743194025,1044554014702,
%T 80967658322673,7236647136567861,737470098999168640,
%U 84879860776191764271,10943491685936397689965,1569258830662933925039980,248708981505469070789015751,43323893019300876864736656191
%N a(n) = ( Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*Bell(k) )*( Sum_{k=1..n} (k-1)!*binomial(n-1, k-1)*binomial(n, k-1) ).
%C Arises in combinatorial field theory.
%D P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).
%D P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G E. H. Duchamp, Combinatorial field theories via boson normal ordering, preprint, Apr 27 2004.
%H Vincenzo Librandi, <a href="/A100524/b100524.txt">Table of n, a(n) for n = 1..280</a>
%H P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, <a href="http://arXiv.org/abs/quant-ph/0405103">Combinatorial field theories via boson normal ordering</a>
%F a(n) = ( Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*Bell(k) )*( Sum_{k=1..n} (k-1)!*binomial(n-1, k-1)*binomial(n, k-1) ).
%F a(n) = A000296(n)*A000262(n).
%p with(combinat): A:=n->add((-1)^(n-k)*binomial(n, k)*bell(k), k=0..n)*add((k-1)!*binomial(n-1, k-1)*binomial(n, k-1), k=1..n): seq(A(n), n=1..18);
%t a[n_]:= Sum[(-1)^(n-k) Binomial[n,k] BellB[k], {k,0,n}] Sum[(k-1)! Binomial[n-1, k-1] Binomial[n, k-1], {k,n}];
%t Table[a[n], {n,20}] (* _Jean-François Alcover_, Nov 11 2018 *)
%o (Magma)
%o F:= Factorial;
%o A000262:= func< n | F(n)*(&+[Binomial(n-1, k)/F(k+1): k in [0..n-1]]) >;
%o A000296:= func< n | (&+[(-1)^(n-k)*Binomial(n, k)*Bell(k): k in [0..n]]) >;
%o A100524:= func< n | A000262(n)*A000296(n) >;
%o [A100524(n): n in [1..30]]; // _G. C. Greubel_, Jun 27 2022
%o (SageMath)
%o def A100524(n): return ( sum((-1)^(n-k)*binomial(n, k)*bell_number(k) for k in (0..n)) )*factorial(n-1)*gen_laguerre(n-1,1,-1)
%o [A100524(n) for n in (1..30)] # _G. C. Greubel_, Jun 27 2022
%Y Cf. A000262, A000296.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Nov 24 2004
|
