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A059340 Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.

%I A059340 #22 Mar 07 2020 08:50:22
%S A059340 1,2,1,5,5,1,15,23,10,1,52,109,76,19,1,203,544,531,224,36,1,877,2876,
%T A059340 3641,2204,631,69,1,4140,16113,25208,20089,8471,1749,134,1,21147,
%U A059340 95495,178564,177631,100171,31331,4838,263,1
%N A059340 Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.
%C A059340 Essentially triangle given by [1,1,1,2,1,3,1,4,1,5,1,6,...] DELTA [0,1,0,2,0,3,0,4,0,5,0,6,...] = [1;1,0;2,1,0;5,5,1,0;15,23,10,1,0;...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Nov 20 2006
%H A059340 G. C. Greubel, <a href="/A059340/b059340.txt">Table of n, a(n) for n = 1..1275</a>
%F A059340 T(n,k) = Sum_{i=0..n} stirling2(n, n-i)*binomial(i, k).
%F A059340 T(n,k) = Sum_{i=0..n} stirling2(n, i)*binomial(n-i, k). - _Peter Luschny_, Aug 06 2015
%e A059340 Triangle starts:
%e A059340    1;
%e A059340    2,   1;
%e A059340    5,   5,   1;
%e A059340   15,  23,  10,   1;
%e A059340   52, 109,  76,  19,   1;
%t A059340 Table[Sum[StirlingS2[n, j]*Binomial[n - j, k], {j, 0, n}], {n, 1,
%t A059340   5}, {k, 0, n - 1}] (* _G. C. Greubel_, Jan 07 2017 *)
%o A059340 (Sage)
%o A059340 T = lambda n,k: sum(stirling_number2(n,j)*binomial(n-j,k) for j in (0..n))
%o A059340 # Also "for n in (0..11): print([T(n,k) for k in (0..n)])" makes sense.
%o A059340 for n in (1..11): print([T(n,k) for k in (0..n-1)]) # _Peter Luschny_, Aug 06 2015
%Y A059340 Row sums = A004211, T(n,0) = A000110.
%K A059340 easy,nonn,tabl
%O A059340 1,2
%A A059340 _Vladeta Jovovic_, Jan 27 2001