%I #17 Jun 30 2023 03:47:57

%S 1,1,1,1,2,1,1,3,5,1,1,4,11,15,1,1,5,19,49,52,1,1,6,29,109,257,203,1,

%T 1,7,41,201,742,1539,877,1,1,8,55,331,1657,5815,10299,4140,1,1,9,71,

%U 505,3176,15821,51193,75905,21147,1,1,10,89,729,5497,35451,170389,498118,609441,115975,1

%N Square array read by antidiagonals upwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

%H Adalbert Kerber, <a href="https://doi.org/10.1016/0012-365X(78)90163-2">A matrix of combinatorial numbers related to the symmetric groups</a>, Discrete Math., 21 (1978), 319-321.

%H Adalbert Kerber, <a href="/A004211/a004211.pdf">A matrix of combinatorial numbers related to the symmetric groups</a>, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]

%e Array begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...

%e 1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, ...

%e 1, 4, 19, 109, 742, 5815, 51193, 498118, 5296321, 60987817, 754940848, 9983845261, ...

%e 1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, ...

%e 1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, ...

%e 1, 7, 55, 505, 5497, 69823, 1007407, 16157905, 284214097, 5432922775, 112034017735, 2476196276617, ...

%e 1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, ...

%e ...

%p with(combinat):

%p T:=(n,k)->add(n^(k-j)*stirling2(k,j),j=1..k);

%p r:=n->[seq(T(n,k),k=1..12)];

%p for n from 0 to 8 do lprint(r(n)); od:

%Y Three versions of this array are A111673, A241578, A241579.

%Y Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_, Apr 29 2014