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a(n) = Sum_{k=0..n} Stirling1(n,k)^2.
4

%I #26 Sep 08 2022 08:44:57

%S 1,1,2,14,194,4402,147552,6838764,418389078,32639603798,3161107700156,

%T 372023906062756,52280302234036252,8645773770675973804,

%U 1661888635268695003484,367390786215560629372920,92552610850186107484661670,26356304249588730696338349990

%N a(n) = Sum_{k=0..n} Stirling1(n,k)^2.

%H Vincenzo Librandi and Vaclav Kotesovec, <a href="/A047796/b047796.txt">Table of n, a(n) for n = 0..250</a> (terms 0..41 from Vincenzo Librandi)

%p seq(add(stirling1(n, k)^2, k = 0..n), n = 0..20); # _G. C. Greubel_, Aug 07 2019

%t Table[Sum[StirlingS1[n,k]^2,{k,0,n}],{n,0,20}] (* _Emanuele Munarini_, Jul 04 2011 *)

%o (Maxima) makelist(sum(stirling1(n,k)^2,k,0,n),n,0,24); \\ _Emanuele Munarini_, Jul 04 2011

%o (PARI) a(n) = sum(k=0, n, stirling(n, k, 1)^2); \\ _Michel Marcus_, Mar 26 2016

%o (Magma) [(&+[StirlingFirst(n,k)^2: k in [0..n]]): n in [0..10]]; // _G. C. Greubel_, Aug 07 2019

%o (Sage) [sum(stirling_number1(n,k)^2 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)^2 )); # _G. C. Greubel_, Aug 07 2019

%Y Cf. A000275, A047797, A342111.

%K nonn

%O 0,3

%A _N. J. A. Sloane_