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A156529 Triangle, T(n, k) = A008517(n+1, k+1)*A008517(n+1, n-k+1), read by rows.

%I A156529 #7 Sep 08 2022 08:45:41
%S A156529 1,2,2,6,64,6,24,1276,1276,24,120,23088,107584,23088,120,720,422712,
%T A156529 6388800,6388800,422712,720,5040,8156160,326165400,1031694400,
%U A156529 326165400,8156160,5040,40320,168521184,15666814800,126099116000,126099116000,15666814800,168521184,40320
%N A156529 Triangle, T(n, k) = A008517(n+1, k+1)*A008517(n+1, n-k+1), read by rows.
%H A156529 G. C. Greubel, <a href="/A156529/b156529.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A156529 T(n, k) = A008517(n+1, k+1)*A008517(n+1, n-k+1).
%F A156529 From _G. C. Greubel_, Dec 30 2021: (Start)
%F A156529 T(n, n-k) = T(n, k).
%F A156529 T(n, 0) = n!. (End)
%e A156529 Triangle begins as:
%e A156529      1;
%e A156529      2,       2;
%e A156529      6,      64,         6;
%e A156529     24,    1276,      1276,         24;
%e A156529    120,   23088,    107584,      23088,       120;
%e A156529    720,  422712,   6388800,    6388800,    422712,     720;
%e A156529   5040, 8156160, 326165400, 1031694400, 326165400, 8156160, 5040;
%t A156529 f[n_, k_]:= f[n, k]= If[k<0 || k>n, 0, If[k==0, 1, (k+1)*f[n-1, k] + (2*n-k+1)*f[n-1, k-1] ]]; (* f = A008517 *)
%t A156529 T[n_, k_]:= f[n+1, k+1]*f[n+1, n-k+1];
%t A156529 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Dec 30 2021 *)
%o A156529 (Magma)
%o A156529 A008517:= func< n,k | (&+[ (-1)^(n+j)*Binomial(2*n+1, j)*StiringFirst(2*n-k-j+1, n-k-j+1) : j in [0..n-k]]) >;
%o A156529 A156529:= func< n,k | A008517(n+1,k+1)*A008517(n+1,n-k+1) >;
%o A156529 [A156529(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 30 2021
%o A156529 (Sage)
%o A156529 @CachedFunction
%o A156529 def A008517(n,k): return sum( (-1)^(n+j)*binomial(2*n+1, j)*stirling_number1(2*n-k-j+1, n-k-j+1)  for j in (0..n-k) )
%o A156529 def A156529(n,k): return A008517(n+1, k+1)*A008517(n+1, n-k+1)
%o A156529 flatten([[A156529(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Dec 30 2021
%Y A156529 Cf. A008517.
%K A156529 nonn,tabl
%O A156529 0,2
%A A156529 _Roger L. Bagula_, Feb 09 2009
%E A156529 Edited by _G. C. Greubel_, Dec 30 2021