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 A361781 #28 Jun 13 2024 01:48:57
%S A361781 1,1,1,1,0,2,1,-1,1,5,1,-2,2,1,15,1,-3,5,-3,4,52,1,-4,10,-13,7,11,203,
%T A361781 1,-5,17,-35,36,-10,41,877,1,-6,26,-75,127,-101,31,162,4140,1,-7,37,
%U A361781 -139,340,-472,293,-21,715,21147,1,-8,50,-233,759,-1573,1787,-848,204,3425,115975
%N A361781 A(n,k) is the n-th term of the k-th inverse binomial transform of the Bell numbers (A000110); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A361781 Alois P. Heinz, <a href="/A361781/b361781.txt">Antidiagonals n = 0..150, flattened</a>
%F A361781 E.g.f. of column k: exp(exp(x) - k*x - 1).
%F A361781 A(n,k) = Sum_{j=0..n} (-k)^j*binomial(n,j)*Bell(n-j).
%e A361781 Square array A(n,k) begins:
%e A361781 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A361781 1, 0, -1, -2, -3, -4, -5, -6, ...
%e A361781 2, 1, 2, 5, 10, 17, 26, 37, ...
%e A361781 5, 1, -3, -13, -35, -75, -139, -233, ...
%e A361781 15, 4, 7, 36, 127, 340, 759, 1492, ...
%e A361781 52, 11, -10, -101, -472, -1573, -4214, -9685, ...
%e A361781 203, 41, 31, 293, 1787, 7393, 23711, 63581, ...
%e A361781 877, 162, -21, -848, -6855, -35178, -134873, -421356, ...
%p A361781 A:= proc(n, k) option remember; uses combinat;
%p A361781 add(binomial(n, j)*(-k)^j*bell(n-j), j=0..n)
%p A361781 end:
%p A361781 seq(seq(A(n, d-n), n=0..d), d=0..10);
%p A361781 # second Maple program:
%p A361781 b:= proc(n, m) option remember;
%p A361781 `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
%p A361781 end:
%p A361781 A:= (n, k)-> b(n, -k):
%p A361781 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A361781 T[n_, k_]:= T[n, k]= If[k==0, BellB[n], Sum[(-k)^j*Binomial[n,j]*BellB[n-j], {j,0,n}]];
%t A361781 A361781[n_, k_]= T[k, n-k];
%t A361781 Table[A361781[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 12 2024 *)
%o A361781 (Magma)
%o A361781 T:= func< n,k | (&+[(-k)^j*Binomial(n,j)*Bell(n-j): j in [0..n]]) >;
%o A361781 A361781:= func< n,k | T(k, n-k) >;
%o A361781 [A361781(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 12 2024
%o A361781 (SageMath)
%o A361781 def T(n,k): return sum( (-k)^j*binomial(n,j)*bell_number(n-j) for j in range(n+1))
%o A361781 def A361781(n, k): return T(k, n-k)
%o A361781 flatten([[A361781(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jun 12 2024
%Y A361781 Columns k=0-5 give: A000110, A000296, A126617, A346738, A346739, A346740.
%Y A361781 Rows n=0-2 give: A000012, A024000, A160457.
%Y A361781 Main diagonal gives A290219.
%Y A361781 Antidiagonal sums give A361380.
%Y A361781 Cf. A108087.
%K A361781 sign,tabl
%O A361781 0,6
%A A361781 _Alois P. Heinz_, Mar 23 2023