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A134980 a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000110(k). 1


%S 1,2,10,77,799,10427,163967,3017562,63625324,1512354975,40012800675,

%T 1166271373797,37134022033885,1282405154139046,47745103281852282,

%U 1906411492286148245,81267367663098939459,3683790958912910588623,176937226305157687076779

%N a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000110(k).

%C Main diagonal of array "The first r-Bell numbers" from r=0 and from n=0, p.3 of Mezo. First 7 rows and columns of array shown. [_Jonathan Vos Post_, Sep 25 2009]

%H Istvan Mezo, <a href="http://arxiv.org/abs/0909.4417">The r-Bell numbers</a>, Sep 24, 2008. [_Jonathan Vos Post_, Sep 25 2009]

%F a(n) = exp(-1)*Sum_{k>=0} (n+k)^n/k!.

%F E.g.f.: A(x) = exp(-1)*Sum_{k>=0} (1+k*x)^k/k!.

%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(n+k). [_Vladeta Jovovic_, Nov 08 2009]

%e From _Jonathan Vos Post_, Sep 25 2009: (Start)

%e The array begins:

%e ===================================================

%e .......|n=0|n=1|.n=2|.n=3.|..n=4.|...n=5.|....n=6.|

%e r=0....|.1.|.1.|..2.|...5.|...15.|....52.|....203.|

%e r=1....|.1.|.2.|..5.|..15.|...52.|...203.|....877.|

%e r=2....|.1.|.3.|.10.|..37.|..151.|...674.|...3263.|

%e r=3....|.1.|.4.|.17.|..77.|..372.|..1915.|..10481.|

%e r=4....|.1.|.5.|.26.|.141.|..799.|..4736.|..29371.|

%e r=5....|.1.|.6.|.37.|.235.|.1540.|.10427.|..73013.|

%e r=6....|.1.|.7.|.50.|.365.|.2727.|.20878.|.163967.|

%e ===================================================

%e (End)

%p with(combinat): a := n -> add(binomial(n, k)*n^(n-k)*bell(k), k=0..n):

%p 1, seq(a(n), n=1..20); # _Emeric Deutsch_, Mar 02 2008

%o (Sage)

%o def A134980(n):

%o return add(binomial(n, k)*n^(n-k)*bell_number(k) for k in (0..n))

%o [A134980(n) for n in (0..18)] # _Peter Luschny_, May 05 2013

%K easy,nonn

%O 0,2

%A _Vladeta Jovovic_, Feb 04 2008

%E More terms from _Emeric Deutsch_, Mar 02 2008

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Last modified November 30 22:52 EST 2015. Contains 264677 sequences.