A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 0, 1, 2, 0, 1, 5, 4, 0, 1, 10, 19, 8, 0, 1, 17, 69, 65, 16, 0, 1, 28, 188, 410, 211, 32, 0, 1, 41, 496, 1726, 2261, 665, 64, 0, 1, 58, 1029, 7182, 14343, 11970, 2059, 128, 0, 1, 77, 2015, 20559, 93345, 112371, 61909, 6305, 256, 0, 1, 100, 3478, 54814, 360612, 1139166, 848506, 315850, 19171, 512, 0
Offset: 0
Examples
A(1,3) = 10 = 5 + 3 + 2, sum of encoded compositions [0,0,1], [0,1,0], [1,0,0]. A(4,2) = 211 = 81 + 54 + 36 + 24 + 16, sum of encoded compositions [0,4], [1,3], [2,2], [3,1], [4,0]. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 2, 5, 10, 17, 28, 41, ... 0, 4, 19, 69, 188, 496, 1029, ... 0, 8, 65, 410, 1726, 7182, 20559, ... 0, 16, 211, 2261, 14343, 93345, 360612, ... 0, 32, 665, 11970, 112371, 1139166, 5827122, ... 0, 64, 2059, 61909, 848506, 13379332, 89131918, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
Crossrefs
Programs
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Maple
A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0, add(ithprime(k)^i*A(n-i, k-1), i=0..n))) end: seq(seq(A(n, d-n), n=0..d), d=0..10); # second Maple program: A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0, ithprime(k)*A(n-1, k)+A(n, k-1))) end: seq(seq(A(n, d-n), n=0..d), d=0..10); -
Mathematica
A[n_, k_] := A[n, k] = If[n == 0, 1, If[k == 0, 0, Prime[k] A[n-1, k] + A[n, k-1]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 06 2021, after 2nd Maple program *)
Formula
A(n,k) = [x^n] Product_{i=1..k} 1/(1-prime(i)*x).
A(n,k) = A124960(n+k,k) for k >= 1.