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) ).

0, 3, 13, 292, 5511, 166091, 6096546, 281962395, 15743194025, 1044554014702, 80967658322673, 7236647136567861, 737470098999168640, 84879860776191764271, 10943491685936397689965, 1569258830662933925039980, 248708981505469070789015751, 43323893019300876864736656191

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