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 A361950 #12 Apr 01 2023 15:48:57
%S A361950 1,1,0,1,1,0,1,2,1,0,1,3,6,1,0,1,4,13,26,1,0,1,5,22,81,162,1,0,1,6,33,
%T A361950 166,721,1442,1,0,1,7,46,287,1726,9153,18306,1,0,1,8,61,450,3309,
%U A361950 24814,165313,330626,1,0,1,9,78,661,5650,50975,494902,4244481,8488962,1,0
%N A361950 Array read by antidiagonals: T(n,k) = n! * Sum_{s} 2^(Sum_{i=1..k-1} s(i)*s(i+1))/(Product_{i=1..k} s(i)!) where the sum is over all nonnegative compositions s of n into k parts.
%C A361950 T(n,k) corresponds to c(k,n) in the Klarner reference. This is an intermediate step in the computation of the number of labeled weakly graded (ranked) posets. The number of elements in the poset is n and the rank k.
%H A361950 Andrew Howroyd, <a href="/A361950/b361950.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%H A361950 D. A. Klarner, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80100-6">The number of graded partially ordered sets</a>, J. Combin. Theory, 6 (1969), 12-19.
%e A361950 Array begins:
%e A361950 ======================================================
%e A361950 n/k| 0 1 2 3 4 5 6 ...
%e A361950 ---+--------------------------------------------------
%e A361950 0 | 1 1 1 1 1 1 1 ...
%e A361950 1 | 0 1 2 3 4 5 6 ...
%e A361950 2 | 0 1 6 13 22 33 46 ...
%e A361950 3 | 0 1 26 81 166 287 450 ...
%e A361950 4 | 0 1 162 721 1726 3309 5650 ...
%e A361950 5 | 0 1 1442 9153 24814 50975 91866 ...
%e A361950 6 | 0 1 18306 165313 494902 1058493 1957066 ...
%e A361950 7 | 0 1 330626 4244481 13729846 29885567 55363650 ...
%e A361950 ...
%e A361950 T(3,2) = 26: the nonnegative integer compositions of 3 with 2 parts are (0,3), (1,2), (2,1), (3,0). These contribute, respectively 2^0*3!/(0!*3!) = 1, 2^2*3!/(1!*2!) = 12, 2^2*3!/(2!*1!) = 12, 2^0*3!/(0!*3!) = 1, so T(3,2) = 1 + 12 + 12 + 1 = 26.
%o A361950 (PARI)
%o A361950 S(M)={matrix(#M, #M, i, j, sum(k=0, i-j, 2^((j-1)*k)*M[i-j+1,k+1])/(j-1)! )}
%o A361950 C(n, m=n)={my(M=matrix(n+1, n+1), c=vector(m+1), A=O(x*x^n)); M[1, 1]=1; c[1]=1+A; for(h=1, m, M=S(M); c[h+1]=sum(i=0, n, vecsum(M[i+1, ])*x^i, A)); c}
%o A361950 R(n)={Mat([Col(serlaplace(p)) | p<-C(n)])}
%o A361950 { my(A=R(6)); for(i=1, #A, print(A[i,])) }
%Y A361950 Columns k=0..7 are A000007, A000012, A047863, A000684, A001827, A001828, A001829, A001830.
%Y A361950 Rows 0..2 are A000012, A000027, A028872(n+1).
%Y A361950 The unlabeled version is A361952.
%Y A361950 Cf. A361951.
%K A361950 nonn,tabl
%O A361950 0,8
%A A361950 _Andrew Howroyd_, Mar 31 2023