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A141686 Triangle read by rows: T(n, k) = binomial(n-1, k-1)*A008292(n, k).

%I A141686 #10 Jan 04 2025 22:47:38
%S A141686 1,1,1,1,8,1,1,33,33,1,1,104,396,104,1,1,285,3020,3020,285,1,1,720,
%T A141686 17865,48320,17865,720,1,1,1729,90153,546665,546665,90153,1729,1,1,
%U A141686 4016,409024,4941104,10933300,4941104,409024,4016,1,1,9117,1722240,38236128,165104604,165104604,38236128,1722240,9117,1
%N A141686 Triangle read by rows: T(n, k) = binomial(n-1, k-1)*A008292(n, k).
%H A141686 Reinhard Zumkeller, <a href="/A141686/b141686.txt">Rows n = 1..125 of table, flattened</a>
%F A141686 T(n, k) = T(n, n-k+1).
%F A141686 Sum_{k=1..n} T(n, k) = A104098(n) (row sums).
%e A141686 Triangle begins as:
%e A141686   1;
%e A141686   1,    1;
%e A141686   1,    8,       1;
%e A141686   1,   33,      33,        1;
%e A141686   1,  104,     396,      104,         1;
%e A141686   1,  285,    3020,     3020,       285,         1;
%e A141686   1,  720,   17865,    48320,     17865,       720,        1;
%e A141686   1, 1729,   90153,   546665,    546665,     90153,     1729,       1;
%e A141686   1, 4016,  409024,  4941104,  10933300,   4941104,   409024,    4016,    1;
%e A141686   1, 9117, 1722240, 38236128, 165104604, 165104604, 38236128, 1722240, 9117, 1;
%t A141686 (* Recurrence for A008292 *)
%t A141686 f[n_, k_]:= If[k==1||k==n,1, (n-k+1)*f[n-1,k-1] + k*f[n-1,k]];
%t A141686 Table[f[n, k]*Binomial[n-1,k-1], {n,12}, {k,n}]//Flatten
%t A141686 (* Second program *)
%t A141686 Needs["Combinatorica`"];
%t A141686 A141686[n_, k_]:= Binomial[n-1,k-1]*Eulerian[n,k-1];
%t A141686 Table[A141686[n,k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Dec 31 2024 *)
%o A141686 (Haskell)
%o A141686 a141686 n k = a141686_tabl !! (n-1) !! (k-1)
%o A141686 a141686_row n = a141686_tabl !! (n-1)
%o A141686 a141686_tabl = zipWith (zipWith (*)) a007318_tabl a008292_tabl
%o A141686 -- _Reinhard Zumkeller_, Apr 16 2014
%o A141686 (Magma)
%o A141686 A141686:= func< n, k | Binomial(n-1, k-1)*EulerianNumber(n, k-1) >;
%o A141686 [A141686(n, k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Dec 31 2024
%o A141686 (Python) # or SageMath
%o A141686 from sage.combinat.combinat import eulerian_number
%o A141686 def A141686(n,k): return binomial(n-1,k-1)*eulerian_number(n,k-1)
%o A141686 print(flatten([[A141686(n,k) for k in range(1,n+1)] for n in range(1,13)])) # _G. C. Greubel_, Dec 31 2024
%Y A141686 Cf. A007318, A008292, A104098.
%K A141686 nonn,tabl
%O A141686 1,5
%A A141686 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 08 2008
%E A141686 keyword:tabl inserted, indices corrected by the Assoc. Eds. of the OEIS, Jun 30 2010