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A343751
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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.
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5
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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
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OFFSET
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0,5
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LINKS
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FORMULA
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A(n,k) = [x^n] Product_{i=1..k} 1/(1-prime(i)*x).
A(n,k) = A124960(n+k,k) for k >= 1.
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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