A324939
Triangle T(n,k) read by rows in which n-th row lists in increasing order all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i); n>=0, 1<=k<=A011782(n).
Original entry on oeis.org
1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 54, 60, 90, 150, 210, 32, 48, 72, 108, 120, 162, 180, 270, 300, 420, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 216, 240, 324, 360, 486, 540, 600, 810, 840, 900, 1260, 1350, 1500, 1890, 2100, 2250, 2940, 3150, 3750, 4410, 4620, 5250, 6930, 7350, 10290, 11550, 16170, 25410, 30030
Offset: 0
Triangle T(n,k) begins:
1;
2;
4, 6;
8, 12, 18, 30;
16, 24, 36, 54, 60, 90, 150, 210;
32, 48, 72, 108, 120, 162, 180, 270, 300, 420, 450, 630, 750, 1050, 1470, 2310;
...
Last elements of rows give
A002110.
-
b:= n-> `if`(n=0, [[]], [seq(map(x-> [j, x[]], b(n-j))[], j=1..n)]):
T:= n-> sort(map(x-> mul(ithprime(i)^x[i], i=1..nops(x)), b(n)))[]:
seq(T(n), n=0..7);
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.
Original entry on oeis.org
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
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, ...
-
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);
-
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 *)
Showing 1-2 of 2 results.
Comments