A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 14, 5, 0, 1, 5, 20, 39, 34, 7, 0, 1, 6, 30, 84, 129, 74, 11, 0, 1, 7, 42, 155, 356, 399, 166, 15, 0, 1, 8, 56, 258, 805, 1444, 1245, 350, 22, 0, 1, 9, 72, 399, 1590, 4055, 5876, 3783, 746, 30, 0
Offset: 0
Examples
A(2,2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b]. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, 7, ... 0, 2, 6, 12, 20, 30, 42, 56, ... 0, 3, 14, 39, 84, 155, 258, 399, ... 0, 5, 34, 129, 356, 805, 1590, 2849, ... 0, 7, 74, 399, 1444, 4055, 9582, 19999, ... 0, 11, 166, 1245, 5876, 20455, 57786, 140441, ... 0, 15, 350, 3783, 23604, 102455, 347010, 983535, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
- Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Crossrefs
Programs
-
Maple
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))) end: A:= (n, k)-> b(n$2, k): seq(seq(A(n, d-n), n=0..d), d=0..12); -
Mathematica
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; A[n_, k_] := b[n, n, k]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)
Formula
G.f. of column k: Product_{i>=1} 1/(1-k*x^i).
T(n,k) = Sum_{i=0..k} C(k,i) * A255970(n,i).
Comments