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.
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, 456, 6, 1, 1, 27007, 4061871, 7958726, 405408, 1709, 7, 1, 1, 353522, 546809243, 7528988476, 1495388159, 9268549, 6427, 8, 1
Offset: 1
Examples
Array begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 7, 37, 266, 2431, 27007, ... 1, 3, 31, 842, 45296, 4061871, 546809243, ... 1, 4, 121, 18252, 7958726, 7528988476, 13130817809439, ... 1, 5, 456, 405408, 1495388159, 15467641899285, 361207016885536095, ... 1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ... ...
Links
- Seiichi Manyama, Antidiagonals n = 1..50, flattened
Crossrefs
Programs
-
Maple
b := proc(n, i, k) local r; option remember; if n = i then 1; elif i < n then 0; elif n < 1 then 0; else add( binomial(i-1,r)*b(n-1,i-1-r,k), r=0..k); end if; end proc; T:=proc(n,k); add(b(n,i,k),i=0..(k+1)*n); end proc; # Peter Luschny, Apr 26 2011 A144510 := proc(n, k) local m; 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;
-
Mathematica
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 *)
Formula
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).
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