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A227450 Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.

%I A227450 #28 Jan 25 2024 04:39:11
%S A227450 2,2,4,2,12,8,2,28,48,16,2,60,200,160,32,2,124,720,1040,480,64,2,252,
%T A227450 2408,5600,4480,1344,128,2,508,7728,27216,33600,17024,3584,256,2,1020,
%U A227450 24200,124320,222432,169344,59136,9216,512,2,2044,74640,545680,1360800,1460928,752640,192000,23040,1024
%N A227450 Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
%C A227450 T(n,k) is the number of ways to separate {1,2,...,n} into 2 ordered subsets S,T so that the union of S and T = {1,2,...,n} then partition each subset so that the total number of blocks over both subsets is equal to k.
%C A227450 Triangle T(n,k), 1<=k<=n, read by rows, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Sep 23 2013
%C A227450 Also the Bell transform of the constant sequence "a(n) = 2". For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016
%H A227450 Indranil Ghosh, <a href="/A227450/b227450.txt">Rows 1..125, flattened</a>
%F A227450 E.g.f.: A(x,y)^2 where A(x,y) is the e.g.f. for A008277.
%e A227450 2,
%e A227450 2, 4,
%e A227450 2, 12, 8,
%e A227450 2, 28, 48, 16,
%e A227450 2, 60, 200, 160, 32,
%e A227450 2, 124, 720, 1040, 480, 64
%t A227450 nn=8; a=Exp[x]-1; Map[Select[#, #>0&]&, Drop[Range[0,nn]! CoefficientList[Series[Exp[y a]^2, {x,0,nn}], {x,y}], 1]]//Grid
%t A227450 (* or *)
%t A227450 Flatten[Table[StirlingS2[n,k]*2^k,{n,1,10},{k,1,n}]] (* _Indranil Ghosh_, Feb 22 2017 *)
%t A227450 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A227450 B = BellMatrix[2&, rows = 12];
%t A227450 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A227450 (Python)  # The function BellMatrix is defined in A264428.
%o A227450 # Adds (1,0,0,0, ..) as column 0.
%o A227450 BellMatrix(lambda n: 2, 9); # _Peter Luschny_, Jan 29 2016
%Y A227450 Cf. A008277.
%K A227450 nonn,tabl
%O A227450 1,1
%A A227450 _Geoffrey Critzer_, Sep 22 2013