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%I #39 Nov 30 2021 10:52:19
%S 2,2,4,4,12,8,12,44,48,16,48,200,280,160,32,240,1096,1800,1360,480,64,
%T 1440,7056,12992,11760,5600,1344,128,10080,52272,105056,108304,62720,
%U 20608,3584,256,80640,438336,944992,1076544,718368,290304,69888,9216,512
%N Triangle read by rows: T(n,k) = S1(n,k)*2^k, where S1(n,k) is an unsigned Stirling number of the first kind (cf. A008275) (n >= 1, 1 <= k <= n).
%C Row sums are factorial numbers.
%C T(n,k) is the number of cycle-colored n-permutations possessing exactly k cycles; two colors are available. - _Steven Finch_, Nov 19 2021
%C Subtriangle (for 1<=k<=n) of triangle given by [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [2,0,2,0,2,0,2,0,2,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 05 2007
%C Also the Bell transform of the sequence "a(n) = 2*(n+1)!/(n+1)". For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%H Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.
%F E.g.f.: 1/(1-x)^(2y). - _Geoffrey Critzer_, Dec 14 2011
%e Triangle begins:
%e 2
%e 2 4
%e 4 12 8
%e 12 44 48 16
%e 48 200 280 160 32
%e Triangle [0,1,1,2,2,3,3,...] DELTA [2,0,2,0,2,0,2,...], for 0<=k<=n, begins:
%e 1;
%e 0, 2;
%e 0, 2, 4;
%e 0, 4, 12, 8;
%e 0, 12, 44, 48, 16;
%e 0, 48, 200, 280, 160, 32;
%p with(combinat): T:=(n,k)->2^k*abs(stirling1(n,k)): for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form - _Emeric Deutsch_, Jan 05 2007
%p A008275 := proc(n,k) if n = 0 and k = 0 then 1 ; elif n = 0 or k = 0 then 0 ; else A008275(n-1,k-1)-(n-1)*A008275(n-1,k) ; fi ; end ; A125553 := proc(n,k) abs(A008275(n,k)*2^k) ; end ; nmax := 10 ; for n from 1 to nmax do for k from 1 to n do printf("%d, ",A125553(n,k)) ; od ; od ; # _R. J. Mathar_, Jan 12 2007
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> 2*(n+1)!/(n+1), 9); # _Peter Luschny_, Jan 27 2016
%t Flatten[Table[Table[2^k Abs[StirlingS1[n,k]], {k,1,n}], {n,1,8}]] (* _Geoffrey Critzer_, Dec 14 2011 *)
%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t B = BellMatrix[Function[n, 2 (n + 1)!/(n + 1)], rows = 12];
%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%Y Cf. A008275, A208529.
%K nonn,tabl,easy
%O 1,1
%A _N. J. A. Sloane_, Jan 04 2007
%E More terms from _R. J. Mathar_, Jan 12 2007
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