A292741 Number A(n,k) of partitions of n with k sorts of part 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 2, 1, 4, 10, 11, 5, 2, 1, 5, 17, 31, 24, 7, 4, 1, 6, 26, 69, 95, 50, 11, 4, 1, 7, 37, 131, 278, 287, 104, 15, 7, 1, 8, 50, 223, 657, 1114, 865, 212, 22, 8, 1, 9, 65, 351, 1340, 3287, 4460, 2599, 431, 30, 12, 1, 10, 82, 521, 2459, 8042, 16439, 17844, 7804, 870, 42, 14
Offset: 0
Examples
A(1,3) = 3: 1a, 1b, 1c. A(2,3) = 10: 2, 1a1a, 1a1b, 1a1c, 1b1a, 1b1b, 1b1c, 1c1a, 1c1b, 1c1c. A(3,2) = 11: 3, 21a, 21b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, 7, ... 1, 2, 5, 10, 17, 26, 37, 50, ... 1, 3, 11, 31, 69, 131, 223, 351, ... 2, 5, 24, 95, 278, 657, 1340, 2459, ... 2, 7, 50, 287, 1114, 3287, 8042, 17215, ... 4, 11, 104, 865, 4460, 16439, 48256, 120509, ... 4, 15, 212, 2599, 17844, 82199, 289540, 843567, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
Crossrefs
Programs
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Maple
b:= proc(n, i, k) option remember; `if`(n=0 or i<2, k^n, add(b(n-i*j, i-1, k), j=0..iquo(n, i))) end: A:= (n, k)-> b(n$2, k): seq(seq(A(n, d-n), n=0..d), d=0..14); -
Mathematica
b[0, , ] = 1; b[n_, i_, k_] := b[n, i, k] = If[i < 2, k^n, Sum[b[n - i*j, i - 1, k], {j, 0, Quotient[n, i]}]]; A[n_, k_] := b[n, n, k]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 19 2018, translated from Maple *)
Formula
G.f. of column k: 1/(1-k*x) * 1/Product_{j>=2} (1-x^j).
A(n,k) = Sum_{j=0..n} A002865(j) * k^(n-j).