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A294032 Triangle read by rows, T(n, k) = Pochhammer(3, k)*Stirling2(3 + n, 3 + k) for n >= 0 and 0 <= k <= n.

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