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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. 2

%I

%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

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Last modified September 26 12:07 EDT 2020. Contains 337371 sequences. (Running on oeis4.)