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Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.
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%I #22 Mar 07 2020 08:50:22

%S 1,2,1,5,5,1,15,23,10,1,52,109,76,19,1,203,544,531,224,36,1,877,2876,

%T 3641,2204,631,69,1,4140,16113,25208,20089,8471,1749,134,1,21147,

%U 95495,178564,177631,100171,31331,4838,263,1

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

%C 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 G. C. Greubel, <a href="/A059340/b059340.txt">Table of n, a(n) for n = 1..1275</a>

%F T(n,k) = Sum_{i=0..n} stirling2(n, n-i)*binomial(i, k).

%F T(n,k) = Sum_{i=0..n} stirling2(n, i)*binomial(n-i, k). - _Peter Luschny_, Aug 06 2015

%e Triangle starts:

%e 1;

%e 2, 1;

%e 5, 5, 1;

%e 15, 23, 10, 1;

%e 52, 109, 76, 19, 1;

%t Table[Sum[StirlingS2[n, j]*Binomial[n - j, k], {j, 0, n}], {n, 1,

%t 5}, {k, 0, n - 1}] (* _G. C. Greubel_, Jan 07 2017 *)

%o (Sage)

%o T = lambda n,k: sum(stirling_number2(n,j)*binomial(n-j,k) for j in (0..n))

%o # Also "for n in (0..11): print([T(n,k) for k in (0..n)])" makes sense.

%o for n in (1..11): print([T(n,k) for k in (0..n-1)]) # _Peter Luschny_, Aug 06 2015

%Y Row sums = A004211, T(n,0) = A000110.

%K easy,nonn,tabl

%O 1,2

%A _Vladeta Jovovic_, Jan 27 2001