|
%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
|
