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a(n) = sum(stirling2(n,k)*stirling2(n+1,k+1),k=0..n).
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%I #8 Dec 15 2023 14:35:58

%S 1,1,4,26,251,3157,50310,978315,22616102,610543614,18953178234,

%T 668200802513,26484030901784,1169889631517219,57168091338306720,

%U 3070545537985858612,180251920018830890897,11507450028966272232867,795397552661209049095698

%N a(n) = sum(stirling2(n,k)*stirling2(n+1,k+1),k=0..n).

%H Vincenzo Librandi, <a href="/A192546/b192546.txt">Table of n, a(n) for n = 0..52</a>

%t Table[Sum[StirlingS2[n,k]StirlingS2[n+1,k+1],{k,0,n}],{n,0,100}]

%t Total/@Table[StirlingS2[n,k]StirlingS2[n+1,k+1],{n,0,20},{k,0,n}] (* _Harvey P. Dale_, Dec 15 2023 *)

%o (Maxima) makelist(sum(stirling2(n,k)*stirling2(n+1,k+1),k,0,n),n,0,24);

%Y Cf. A047793

%K nonn

%O 0,3

%A _Emanuele Munarini_, Jul 04 2011