login
a(n) = Sum_{k=0..n} C(n,k)*|Stirling1(n,k)*Stirling2(n,k)|.
4

%I #11 Sep 08 2022 08:44:57

%S 1,1,3,34,631,16871,617356,28968990,1680536159,117572734195,

%T 9715771690081,932711356031016,102653506699902874,

%U 12810868034079756421,1795954763065584594656,280569433733767673934426,48506369621902094002862671,9224242346164172284054561019

%N a(n) = Sum_{k=0..n} C(n,k)*|Stirling1(n,k)*Stirling2(n,k)|.

%H G. C. Greubel, <a href="/A047794/b047794.txt">Table of n, a(n) for n = 0..275</a>

%p seq(add((-1)^(n-k)*binomial(n, k)*stirling1(n, k)*stirling2(n, k), k = 0 .. n), n = 0..20); # _G. C. Greubel_, Aug 07 2019

%t Table[Sum[Binomial[n,k]Abs[StirlingS1[n,k]StirlingS2[n,k]],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Apr 10 2012 *)

%o (PARI) {a(n) = sum(k=0,n, (-1)^(n-k)*stirling(n,k,1)*stirling(n,k,2) *binomial(n,k))};

%o vector(20, n, n--; a(n)) \\ _G. C. Greubel_, Aug 07 2019

%o (Magma) [(&+[(-1)^(n-k)*StirlingFirst(n,k)*StirlingSecond(n,k) *Binomial(n,k): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 07 2019

%o (Sage) [sum(stirling_number1(n,k)*stirling_number2(n,k)*binomial(n,k) for k in (0..n)) for n in (0..20)] # _G. C. Greubel_, Aug 07 2019

%o (GAP) List([0..20], n-> Sum([0..n], k-> Stirling1(n,k)*Stirling2(n,k) *Binomial(n,k) )); # _G. C. Greubel_, Aug 07 2019

%Y Cf. A008275, A008277, A047792, A047793, A047795.

%K nonn

%O 0,3

%A _N. J. A. Sloane_