%I #12 Apr 01 2023 15:48:57
%S 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 166,721,1442,1,0,1,7,46,287,1726,9153,18306,1,0,1,8,61,450,3309,
%U 24814,165313,330626,1,0,1,9,78,661,5650,50975,494902,4244481,8488962,1,0
%N 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 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 Andrew Howroyd, <a href="/A361950/b361950.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%H 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 Array begins:
%e ======================================================
%e n/k| 0 1 2 3 4 5 6 ...
%e ---+--------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 ...
%e 1 | 0 1 2 3 4 5 6 ...
%e 2 | 0 1 6 13 22 33 46 ...
%e 3 | 0 1 26 81 166 287 450 ...
%e 4 | 0 1 162 721 1726 3309 5650 ...
%e 5 | 0 1 1442 9153 24814 50975 91866 ...
%e 6 | 0 1 18306 165313 494902 1058493 1957066 ...
%e 7 | 0 1 330626 4244481 13729846 29885567 55363650 ...
%e ...
%e 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 (PARI)
%o 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 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 R(n)={Mat([Col(serlaplace(p)) | p<-C(n)])}
%o { my(A=R(6)); for(i=1, #A, print(A[i,])) }
%Y Columns k=0..7 are A000007, A000012, A047863, A000684, A001827, A001828, A001829, A001830.
%Y Rows 0..2 are A000012, A000027, A028872(n+1).
%Y The unlabeled version is A361952.
%Y Cf. A361951.
%K nonn,tabl
%O 0,8
%A _Andrew Howroyd_, Mar 31 2023