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A348296
Irregular table T(n, k), n > 0, k = 1..A000120(n), read by rows; the n-th contains, in ascending order, the distinct powers of 2 summing to n.
6
1, 2, 1, 2, 4, 1, 4, 2, 4, 1, 2, 4, 8, 1, 8, 2, 8, 1, 2, 8, 4, 8, 1, 4, 8, 2, 4, 8, 1, 2, 4, 8, 16, 1, 16, 2, 16, 1, 2, 16, 4, 16, 1, 4, 16, 2, 4, 16, 1, 2, 4, 16, 8, 16, 1, 8, 16, 2, 8, 16, 1, 2, 8, 16, 4, 8, 16, 1, 4, 8, 16, 2, 4, 8, 16, 1, 2, 4, 8, 16, 32
OFFSET
1,2
FORMULA
T(n, k) = 2^A133457(n, k).
T(n, 1) = A006519(n).
T(n, A000120(n)) = A053644(n).
Sum_{k = 1..A000120(n)} T(n, k) = n.
Sum_{k = 1..A000120(n)} T(n, k) * (-1)^(k-1) = A065620(n).
Product_{k = 1..A000120(n)} T(n, k) = A059867(n).
T(2*n, k) = 2*T(n, k).
EXAMPLE
Triangle T(n, k) begins:
n n-th row
-- ------------
1 [1]
2 [2]
3 [1, 2]
4 [4]
5 [1, 4]
6 [2, 4]
7 [1, 2, 4]
8 [8]
9 [1, 8]
10 [2, 8]
11 [1, 2, 8]
12 [4, 8]
13 [1, 4, 8]
14 [2, 4, 8]
15 [1, 2, 4, 8]
MATHEMATICA
Array[DeleteCases[Union@ NumberExpand[#, 2], 0] &, 32] // Flatten (* Michael De Vlieger, Jul 19 2022 *)
PROG
(PARI) row(n) = { my (r=vector(hammingweight(n))); for (k=1, #r, n -= r[k] = 2^valuation(n, 2)); return (r) }
KEYWORD
nonn,tabf,base,easy
AUTHOR
Rémy Sigrist, Jul 18 2022
STATUS
approved