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A046817 Triangle of generalized Stirling numbers of 2nd kind.

%I A046817 #33 Jan 05 2025 19:51:35
%S A046817 1,1,2,1,6,5,1,12,32,15,1,20,110,175,52,1,30,280,945,1012,203,1,42,
%T A046817 595,3465,8092,6230,877,1,56,1120,10010,40992,70756,40819,4140,1,72,
%U A046817 1932,24570,156072,479976,638423,283944,21147,1,90,3120,53550,487704,2350950,5660615,5971350
%N A046817 Triangle of generalized Stirling numbers of 2nd kind.
%H A046817 Tilman Piesk, <a href="/A046817/b046817.txt">First 100 rows, flattened</a>
%H A046817 R. Fray, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-4/fray.pdf">A generating function associated with the generalized Stirling numbers</a>, Fib. Quart. 5 (1967), 356-366.
%F A046817 a(n, k) = Sum_{i=k..n} S2(n, i)*S2(i, k).
%F A046817 E.g.f.: exp(exp(exp(x*y)-1)-1)^(1/y). - _Vladeta Jovovic_, Dec 14 2003
%e A046817 Triangle begins:
%e A046817       k = 0    1    2    3    4          sum
%e A046817 n
%e A046817 1         1                                1
%e A046817 2         1    2                           3
%e A046817 3         1    6    5                     12
%e A046817 4         1   12   32   15                60
%e A046817 5         1   20  110  175   52          358
%t A046817 a[n_, k_] = Sum[StirlingS2[n, i]*StirlingS2[i, k], {i, k, n}]; Flatten[Table[a[n, k], {n, 1, 10}, {k, n, 1, -1}]][[1 ;; 53]]  (* _Jean-François Alcover_, Apr 26 2011 *)
%Y A046817 Diagonals give A000558, A000559, A000110, A002378, etc.
%Y A046817 Row sums give A000258.
%Y A046817 Horizontal mirror triangle is A039810 (matrix square of Stirling2).
%K A046817 tabl,nonn,easy,nice
%O A046817 0,3
%A A046817 _N. J. A. Sloane_
%E A046817 More terms from _David W. Wilson_, Jan 13 2000