This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A142070 #22 Apr 25 2024 09:23:13
%S A142070 1,-1,2,2,-7,6,-6,29,-46,24,24,-146,329,-326,120,-120,874,-2521,3604,
%T A142070 -2556,720,720,-6084,21244,-39271,40564,-22212,5040,-5040,48348,
%U A142070 -197380,444849,-598116,479996,-212976,40320,40320,-432144,2014172,-5335212,8788569,-9223012,6023772,-2239344,362880
%N A142070 Triangle T(n,k) read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (i+1)*x-i in row n>=0 and column 0<=k<=n.
%C A142070 This is essentially a signed version of A088996. - _Peter Bala_, Jan 09 2017
%H A142070 G. C. Greubel, <a href="/A142070/b142070.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A142070 T(n, k) = [x^k]( Product_{j=1..n} ((1+j)*x - j) ).
%F A142070 Sum_{k=0..n} T(n, k) = 1.
%F A142070 From _G. C. Greubel_, Feb 24 2022: (Start)
%F A142070 T(n, k) = (-1)^(n-k) * Sum_{j=0..n} (-1)^j*binomial(j,n-k)*Stirling1(n+1, n-j+1).
%F A142070 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(j+n-k,n-k)*Stirling1(n+1, k-j+1).
%F A142070 T(n, 0) = (-1)^n * n!.
%F A142070 T(n, n) = (n+1)!. (End)
%e A142070 Triangle begins as:
%e A142070 1;
%e A142070 -1, 2;
%e A142070 2, -7, 6;
%e A142070 -6, 29, -46, 24;
%e A142070 24, -146, 329, -326, 120;
%e A142070 -120, 874, -2521, 3604, -2556, 720;
%e A142070 720, -6084, 21244, -39271, 40564, -22212, 5040;
%e A142070 -5040, 48348, -197380, 444849, -598116, 479996, -212976, 40320;
%e A142070 40320, -432144, 2014172, -5335212, 8788569, -9223012, 6023772, -2239344, 362880;
%p A142070 A142070 := proc(n,k)
%p A142070 local x,i ;
%p A142070 mul( (i+1)*x-i,i=1..n) ;
%p A142070 expand(%) ;
%p A142070 coeff(%,x,k) ;
%p A142070 end proc:
%t A142070 (* First program *)
%t A142070 p[x_, n_]:= Product[(i+1)*x - i, {i, n}];
%t A142070 Table[CoefficientList[p[x, n], x], {n,0,10}]//Flatten
%t A142070 (* Second program *)
%t A142070 T[n_, k_]:= T[n, k]= Sum[(-1)^j*Binomial[j+n-k, n-k]*StirlingS1[n+1,k-j+1], {j, 0, k}];
%t A142070 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 24 2022 *)
%o A142070 (Magma)
%o A142070 A142070:= func< n,k | (-1)^(n-k)*(&+[(-1)^j*Binomial(j,n-k)*StirlingFirst(n+1,n-j+1): j in [0..n]]) >;
%o A142070 [A142070(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 24 2022
%o A142070 (Sage)
%o A142070 def A142070(n,k): return (-1)^(n-k)*sum(binomial(j+n-k, n-k)*stirling_number1(n+1, k-j+1) for j in (0..k))
%o A142070 flatten([[A142070(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 24 2022
%o A142070 (PARI) row(n) = Vecrev(prod(j=1, n, (1+j)*x - j)); \\ _Michel Marcus_, Feb 24 2022
%Y A142070 Cf. A048994, A088996.
%K A142070 sign,tabl,easy
%O A142070 0,3
%A A142070 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008