A348296 - OEIS

%I #47 Jan 05 2024 14:28:56

%S 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,

%T 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,

%U 16,4,8,16,1,4,8,16,2,4,8,16,1,2,4,8,16,32

%N 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.

%H Michael De Vlieger, <a href="/A348296/b348296.txt">Table of n, a(n) for n = 1..10870</a> (rows n = 1..2000, flattened)

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F T(n, k) = 2^A133457(n, k).

%F T(n, 1) = A006519(n).

%F T(n, A000120(n)) = A053644(n).

%F Sum_{k = 1..A000120(n)} T(n, k) = n.

%F Sum_{k = 1..A000120(n)} T(n, k) * (-1)^(k-1) = A065620(n).

%F Product_{k = 1..A000120(n)} T(n, k) = A059867(n).

%F T(2*n, k) = 2*T(n, k).

%e Triangle T(n, k) begins:

%e   n   n-th row

%e   --  ------------

%e    1  [1]

%e    2  [2]

%e    3  [1, 2]

%e    4  [4]

%e    5  [1, 4]

%e    6  [2, 4]

%e    7  [1, 2, 4]

%e    8  [8]

%e    9  [1, 8]

%e   10  [2, 8]

%e   11  [1, 2, 8]

%e   12  [4, 8]

%e   13  [1, 4, 8]

%e   14  [2, 4, 8]

%e   15  [1, 2, 4, 8]

%t Array[DeleteCases[Union@ NumberExpand[#, 2], 0] &, 32] // Flatten (* _Michael De Vlieger_, Jul 19 2022 *)

%o (PARI) row(n) = { my (r=vector(hammingweight(n))); for (k=1, #r, n -= r[k] = 2^valuation(n, 2)); return (r) }

%Y Cf. A000079, A000120, A006519, A053644, A059867, A065620, A133457.

%K nonn,tabf,base,easy

%O 1,2

%A _Rémy Sigrist_, Jul 18 2022