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A142472 Triangle T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k), read by rows.

%I A142472 #10 Apr 03 2021 03:08:17
%S A142472 1,-4,1,21,-18,1,-140,240,-48,1,1140,-3150,1300,-100,1,-11004,43620,
%T A142472 -29700,4800,-180,1,123074,-650769,647780,-175175,13965,-294,1,
%U A142472 -1566928,10517108,-14190400,5676160,-764400,34496,-448,1,22390488,-184052520,319680732,-175091112,35160048,-2698920,75600,-648,1
%N A142472 Triangle T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k), read by rows.
%C A142472 Row sums are: 1, -3, 4, 53, -809, 7537, -41418, -294411, 15463669, -352665269, ....
%H A142472 G. C. Greubel, <a href="/A142472/b142472.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A142472 T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k).
%e A142472 The triangle begins as:
%e A142472          1;
%e A142472         -4,          1;
%e A142472         21,        -18,         1;
%e A142472       -140,        240,       -48,          1;
%e A142472       1140,      -3150,      1300,       -100,        1;
%e A142472     -11004,      43620,    -29700,       4800,     -180,        1;
%e A142472     123074,    -650769,    647780,    -175175,    13965,     -294,     1;
%e A142472   -1566928,   10517108, -14190400,    5676160,  -764400,    34496,  -448,    1;
%e A142472   22390488, -184052520, 319680732, -175091112, 35160048, -2698920, 75600, -648, 1;
%p A142472 A142472:= (n,k)-> binomial(n,k)*add(Stirling1(n,j)*Stirling1(j,k), j=k..n);
%p A142472 seq(seq(A142472(n,k), k=1..n), n=1..12); # _G. C. Greubel_, Apr 02 2021
%t A142472 T[n_, k_]:= Binomial[n, k]*Sum[StirlingS1[n, j]*StirlingS1[j, k], {j, k, n}];
%t A142472 Table[T[n, k], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Apr 02 2021 *)
%o A142472 (Magma)
%o A142472 A142472:= func< n,k | Binomial(n,k)*(&+[StirlingFirst(n,j)*StirlingFirst(j,k): j in [k..n]]) >;
%o A142472 [A142472(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 02 2021
%o A142472 (Sage)
%o A142472 def A142472(n,k): return (-1)^(n-k)*binomial(n,k)*sum( stirling_number1(n, j)*stirling_number1(j, k) for j in (k..n) )
%o A142472 flatten([[A142472(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 02 2021
%Y A142472 Cf. A008275, A039814.
%K A142472 sign,tabl
%O A142472 1,2
%A A142472 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 22 2008
%E A142472 Edited by _N. J. A. Sloane_, Sep 26 2008