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 A144510 #33 May 19 2019 20:38:42
%S A144510 1,1,1,1,2,1,1,7,3,1,1,37,31,4,1,1,266,842,121,5,1,1,2431,45296,18252,
%T A144510 456,6,1,1,27007,4061871,7958726,405408,1709,7,1,1,353522,546809243,
%U A144510 7528988476,1495388159,9268549,6427,8,1
%N A144510 Array T(n,k) (n >= 1, k >= 0) read by downwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., i] into exactly k nonempty blocks, each of size at most n, for any i in the range n <= i <= k*n.
%H A144510 Seiichi Manyama, <a href="/A144510/b144510.txt">Antidiagonals n = 1..50, flattened</a>
%F A144510 T(n,k) = (1/k!)*Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..n} multinomial(i_1+i_2+...+i_k; i_1, i_2, ..., i_k).
%F A144510 T(n,k) = (1/k!)*Sum_{m=k..k*n} m! [x^m](e^x Gamma(n+1,x)/Gamma(n+1)-1)^k. Here [x^m]f(x) is the coefficient of x^m in the series expansion of f(x). - _Peter Luschny_, Apr 26 2011
%e A144510 Array begins:
%e A144510 1, 1, 1, 1, 1, 1, 1, ...
%e A144510 1, 2, 7, 37, 266, 2431, 27007, ...
%e A144510 1, 3, 31, 842, 45296, 4061871, 546809243, ...
%e A144510 1, 4, 121, 18252, 7958726, 7528988476, 13130817809439, ...
%e A144510 1, 5, 456, 405408, 1495388159, 15467641899285, 361207016885536095, ...
%e A144510 1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ...
%e A144510 ...
%p A144510 b := proc(n, i, k) local r;
%p A144510 option remember;
%p A144510 if n = i then 1;
%p A144510 elif i < n then 0;
%p A144510 elif n < 1 then 0;
%p A144510 else add( binomial(i-1,r)*b(n-1,i-1-r,k), r=0..k);
%p A144510 end if;
%p A144510 end proc;
%p A144510 T:=proc(n,k); add(b(n,i,k),i=0..(k+1)*n); end proc;
%p A144510 # _Peter Luschny_, Apr 26 2011
%p A144510 A144510 := proc(n, k) local m;
%p A144510 add(m!*coeff(expand((exp(x)*GAMMA(n+1,x)/GAMMA(n+1)-1)^k),x,m),m=k..k*n)/k! end: for row from 1 to 6 do seq(A144510(row, col), col = 0..5) od;
%t A144510 multinomial[n_, k_List] := n!/Times @@ (k!); t[n_, k_] := Module[{i, ik}, ik = Array[i, k]; 1/k!* Sum[multinomial[Total[ik], ik], Evaluate[Sequence @@ Thread[{ik, 1, n}]]]]; Table[t[n-k, k], {n, 1, 10}, {k, n-1, 0, -1}] // Flatten (* _Jean-François Alcover_, Jan 14 2014 *)
%Y A144510 For the transposed array see A144512.
%Y A144510 Rows include A001515, A144416, A144508, A144509.
%Y A144510 Columns include A048775, A144511.
%Y A144510 A(n+1,n) gives A281901.
%Y A144510 A(n,n) gives A308296.
%Y A144510 Cf. A308292.
%K A144510 nonn,tabl
%O A144510 1,5
%A A144510 _David Applegate_ and _N. J. A. Sloane_, Dec 15 2008, Jan 30 2009