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 A309010 #61 Aug 30 2022 14:15:53
%S A309010 1,1,2,1,2,3,1,2,4,4,1,2,6,8,5,1,2,10,20,16,6,1,2,18,56,70,32,7,1,2,
%T A309010 34,164,346,252,64,8,1,2,66,488,1810,2252,924,128,9,1,2,130,1460,9826,
%U A309010 21252,15184,3432,256,10,1,2,258,4376,54850,206252,263844,104960,12870,512,11
%N A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.
%C A309010 A(n,k) is the constant term in the expansion of (Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0. - _Seiichi Manyama_, Oct 27 2019
%C A309010 Let B_k be the binomial poset containing all k-tuples of equinumerous subsets of {1,2,...} ordered by inclusion componentwise (described in Stanley reference below). Then A(k,n) is the number of elements in any n-interval of B_k. - _Geoffrey Critzer_, Apr 16 2020
%C A309010 Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - Product_{j=1..k} x_j) for k>0. - _Seiichi Manyama_, Jul 11 2020
%D A309010 R. P. Stanley, Enumerative Combinatorics Vol I, Second Edition, Cambridge, 2011, Example 3.18.3 d, page 366.
%H A309010 Seiichi Manyama, <a href="/A309010/b309010.txt">Antidiagonals n = 0..100, flattened</a>
%F A309010 A(n, k) = Sum_{j=0..n} binomial(n,j)^k (array).
%F A309010 A(n, n+1) = A328812(n).
%F A309010 A(n, n) = A167010(n).
%F A309010 T(n, k) = A(k, n-k) (antidiagonals).
%F A309010 T(n, n) = A000027(n+1).
%F A309010 T(n, n-1) = A000079(n-1).
%F A309010 T(n, n-2) = A000984(n-2).
%F A309010 T(n, n-3) = A000172(n-3).
%F A309010 T(n, n-4) = A005260(n-4).
%F A309010 T(n, n-5) = A005261(n-5).
%F A309010 T(n, n-6) = A069865(n-6).
%F A309010 T(n, n-7) = A182421(n-7).
%F A309010 T(n, n-8) = A182422(n-8).
%F A309010 T(n, n-9) = A182446(n-9).
%F A309010 T(n, n-10) = A182447(n-10).
%F A309010 T(n, n-11) = A342294(n-11).
%F A309010 T(n, n-12) = A342295(n-12).
%F A309010 Sum_{n>=0} A(n,k) x^n/(n!^k) = (Sum_{n>=0} x^n/(n!^k))^2. - _Geoffrey Critzer_, Apr 17 2020
%e A309010 Square array, A(n, k), begins:
%e A309010 1, 1, 1, 1, 1, 1, ... A000012;
%e A309010 2, 2, 2, 2, 2, 2, ... A007395;
%e A309010 3, 4, 6, 10, 18, 34, ... A052548;
%e A309010 4, 8, 20, 56, 164, 488, ... A115099;
%e A309010 5, 16, 70, 346, 1810, 9826, ...
%e A309010 6, 32, 252, 2252, 21252, 206252, ...
%e A309010 Antidiagonals, T(n, k), begin:
%e A309010 1;
%e A309010 1, 2;
%e A309010 1, 2, 3;
%e A309010 1, 2, 4, 4;
%e A309010 1, 2, 6, 8, 5;
%e A309010 1, 2, 10, 20, 16, 6;
%e A309010 1, 2, 18, 56, 70, 32, 7;
%e A309010 1, 2, 34, 164, 346, 252, 64, 8;
%e A309010 1, 2, 66, 488, 1810, 2252, 924, 128, 9;
%e A309010 1, 2, 130, 1460, 9826, 21252, 15184, 3432, 256, 10;
%t A309010 nn = 8; Table[ek[x_] := Sum[x^n/n!^k, {n, 0, nn}];Range[0, nn]!^k CoefficientList[Series[ek[x]^2, {x, 0, nn}],x], {k, 0, nn}] // Transpose // Grid (* _Geoffrey Critzer_, Apr 17 2020 *)
%o A309010 (PARI) A(n, k) = sum(j=0, n, binomial(n, j)^k); \\ _Seiichi Manyama_, Jan 08 2022
%o A309010 (Magma) [(&+[Binomial(k,j)^(n-k): j in [0..k]]): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 26 2022
%o A309010 (SageMath) flatten([[sum(binomial(k,j)^(n-k) for j in (0..k)) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Aug 26 2022
%Y A309010 Columns k=0..12 give A000027(n+1), A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
%Y A309010 Main diagonal gives A167010.
%Y A309010 Cf. A328747, A328748, A328807, A328812.
%Y A309010 Cf. A000012, A007395, A052548, A115099.
%K A309010 nonn,tabl
%O A309010 0,3
%A A309010 _Seiichi Manyama_, Jul 06 2019