%I #18 Jan 15 2020 13:29:44
%S 1,6,3,25,30,12,90,195,180,60,301,1050,1680,1260,360,966,5103,12600,
%T 15960,10080,2520,3025,23310,83412,158760,166320,90720,20160,9330,
%U 102315,510300,1369620,2116800,1890000,907200,181440
%N Triangle read by rows, T(n, k) = Pochhammer(3, k)*Stirling2(3 + n, 3 + k) for n >= 0 and 0 <= k <= n.
%H G. C. Greubel, <a href="/A294032/b294032.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F E.g.f.: (1/2)*exp(x)*(2*y + 9*exp(2*x) + y^2+1-11*exp(3*x)*y + 15*y^2*exp(2*x) - 7*y^2*exp(x) - 13*y^2*exp(3*x) + 4*exp(4*x)*y^2 - 8*exp(x) + 24*y*exp(2*x) - 15*y*exp(x))/(1 - y*(exp(x) - 1))^3.
%F T(n, k) = A293617(3, n, k).
%e Triangle starts:
%e [0] 1
%e [1] 6, 3
%e [2] 25, 30, 12
%e [3] 90, 195, 180, 60
%e [4] 301, 1050, 1680, 1260, 360
%e [5] 966, 5103, 12600, 15960, 10080, 2520
%e [6] 3025, 23310, 83412, 158760, 166320, 90720, 20160
%e [7] 9330, 102315, 510300, 1369620, 2116800, 1890000, 907200, 181440
%p A294032 := (n, k) -> pochhammer(3, k)*Stirling2(n + 3, k + 3):
%p seq(seq(A294032(n, k), k=0..n), n=0..7);
%p T := (n, k) -> A293617(3, n, k): seq(seq(T(n, k), k=0..n), n=0..7);
%t Table[Pochhammer[3, k] StirlingS2[3 + n, 3 + k], {n, 0, 7}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Oct 22 2017 *)
%o (PARI) for(n=0,10, for(k=0,n, print1((k+2)!*stirling(n+3,k+3,2)/2, ", "))) \\ _G. C. Greubel_, Nov 19 2017
%Y T(n, 0) = A000392(n+3), T(n, n) = A001710(n+2).
%Y Row sums A002051(n+3), alternating row sums A000225(n+1).
%Y Cf. A028246 (m=1), A053440 (m=2), this seq. (m=3), A293617 (hub).
%K nonn,tabl
%O 0,2
%A _Peter Luschny_, Oct 22 2017