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A217537 Triangle read by rows, T(n,k) = T(n-1,k-1) + k*T(n-1,k) + (k+1)*T(n-1,k+1), T(0,0) = 1, n >= 0, k >= 0.

%I A217537 #28 Mar 26 2022 16:31:39
%S A217537 1,0,1,1,1,1,1,4,3,1,4,11,13,6,1,11,41,55,35,10,1,41,162,256,200,80,
%T A217537 15,1,162,715,1274,1176,595,161,21,1,715,3425,6791,7182,4361,1526,294,
%U A217537 28,1,3425,17722,38553,45781,32256,13755,3486,498,36,1,17722,98253
%N A217537 Triangle read by rows, T(n,k) = T(n-1,k-1) + k*T(n-1,k) + (k+1)*T(n-1,k+1), T(0,0) = 1, n >= 0, k >= 0.
%C A217537 Related to set partitions without singletons, T(n,0) = A000296(n).
%H A217537 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/AignerTriangles">Aigner Triangles</a>
%F A217537 From _Mélika Tebni_, Mar 26 2022: (Start)
%F A217537 E.g.f. column k: exp(exp(x) - 1 - x)*(exp(x) - 1)^k / k!, k >= 0.
%F A217537 Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n. (End)
%e A217537 [0]    1,
%e A217537 [1]    0,    1,
%e A217537 [2]    1,    1,    1,
%e A217537 [3]    1,    4,    3,    1,
%e A217537 [4]    4,   11,   13,    6,    1,
%e A217537 [5]   11,   41,   55,   35,   10,    1,
%e A217537 [6]   41,  162,  256,  200,   80,   15,    1,
%e A217537 [7]  162,  715, 1274, 1176,  595,  161,   21,    1,
%e A217537 [8]  715, 3425, 6791, 7182, 4361, 1526,  294,   28,    1
%t A217537 T[0, 0] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k*T[n - 1, k] + (k + 1)*T[n - 1, k + 1]; T[_, _] = 0;
%t A217537 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 02 2019 *)
%o A217537 (Sage)
%o A217537 def A217537_triangle(dim):
%o A217537     T = matrix(ZZ,dim,dim)
%o A217537     for n in range(dim): T[n,n] = 1
%o A217537     for n in (1..dim-1):
%o A217537         for k in (0..n-1):
%o A217537             T[n,k] = T[n-1,k-1]+k*T[n-1,k]+(k+1)*T[n-1,k+1]
%o A217537     return T
%o A217537 A217537_triangle(9)
%Y A217537 Row sums are A217924, A000296 (first column).
%K A217537 nonn,tabl
%O A217537 1,8
%A A217537 _Peter Luschny_, Oct 06 2012