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 A100524 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) ). 1
 0, 3, 13, 292, 5511, 166091, 6096546, 281962395, 15743194025, 1044554014702, 80967658322673, 7236647136567861, 737470098999168640, 84879860776191764271, 10943491685936397689965, 1569258830662933925039980, 248708981505469070789015751, 43323893019300876864736656191 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Arises in combinatorial field theory. REFERENCES 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). 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..280 P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering FORMULA 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) ). a(n) = A000296(n)*A000262(n). MAPLE 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); MATHEMATICA 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}]; Table[a[n], {n, 20}] (* Jean-François Alcover, Nov 11 2018 *) PROG (Magma) F:= Factorial; A000262:= func< n | F(n)*(&+[Binomial(n-1, k)/F(k+1): k in [0..n-1]]) >; A000296:= func< n | (&+[(-1)^(n-k)*Binomial(n, k)*Bell(k): k in [0..n]]) >; A100524:= func< n | A000262(n)*A000296(n) >; [A100524(n): n in [1..30]]; // G. C. Greubel, Jun 27 2022 (SageMath) 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) [A100524(n) for n in (1..30)] # G. C. Greubel, Jun 27 2022 CROSSREFS Cf. A000262, A000296. Sequence in context: A042823 A132560 A128385 * A000859 A045748 A113526 Adjacent sequences: A100521 A100522 A100523 * A100525 A100526 A100527 KEYWORD nonn AUTHOR Emeric Deutsch, Nov 24 2004 STATUS approved

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Last modified October 4 15:50 EDT 2023. Contains 365885 sequences. (Running on oeis4.)