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A355811 a(n) is the number at the apex of a triangle whose base contains the distinct powers of 2 summing to n (in ascending order), and each number in a higher row is the product of the two numbers directly below it; a(0) = 1.

%I A355811 #5 Jul 19 2022 08:10:25
%S A355811 1,1,2,2,4,4,8,16,8,8,16,32,32,128,256,4096,16,16,32,64,64,256,512,
%T A355811 8192,128,1024,2048,65536,4096,524288,1048576,4294967296,32,32,64,128,
%U A355811 128,512,1024,16384,256,2048,4096,131072,8192,1048576,2097152,8589934592,512
%N A355811 a(n) is the number at the apex of a triangle whose base contains the distinct powers of 2 summing to n (in ascending order), and each number in a higher row is the product of the two numbers directly below it; a(0) = 1.
%H A355811 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A355811 a(n) = n iff n is a power of 2.
%F A355811 a(2*n) = a(n) * 2^A048896(n-1) for any n > 0.
%e A355811 For n = 27:
%e A355811 - we have the following triangle:
%e A355811            65536
%e A355811          32  2048
%e A355811        2    16   128
%e A355811     1     2     8    16
%e A355811 - so a(27) = 65536.
%o A355811 (PARI) a(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n-=b[k]=2^valuation(n, 2)); while (#b>1, b=vector(#b-1, k, b[k+1]*b[k])); if (#b, b[1], 1) }
%Y A355811 See A355807 for similar sequences.
%Y A355811 Cf. A048896, A348296.
%K A355811 nonn,base
%O A355811 0,3
%A A355811 _Rémy Sigrist_, Jul 18 2022