A227287 - OEIS

%I #27 Jul 06 2013 19:30:16

%S 1,2,2,3,1,3,4,4,1,2,2,3,2,6,7,5,1,2,2,3,1,3,4,4,1,2,2,4,6,12,11,6,1,

%T 2,2,3,1,3,4,4,1,2,2,3,2,6,7,5,1,2,2,3,1,3,4,4,1,2,3,9,16,22,16,7,1,2,

%U 2,3,1,3,4,4,1,2,2,3,2,6,7,5,1,2,2,3,1,3,4,4,1,2,2,4,6,12,11,6,1,2,2,3,1,3,4,4,1,2,2,3,2,6,7,5,1,2,2,3,1,3,4,4,1,3,9,24,36,37,22,8,1

%N G.f.: Sum_{n>=0} x^(n - b(n)) * (1+x)^b(n), where b(n) = A007814(n), which is the exponent of the highest power of 2 dividing n.

%C The g.f. of related sequence A227277 is: Sum_{n>=0} x^n*(1+x)^A007814(n).

%H Paul D. Hanna, <a href="/A227287/b227287.txt">Table of n, a(n) for n = 0..10000</a>

%F Limit of rows, when read in reverse, of A227277 when formatted into a triangle having 2^n terms in row n>=0.

%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 3*x^3 + x^4 + 3*x^5 + 4*x^6 + 4*x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 2*x^12 + 6*x^13 + 7*x^14 + 5*x^15 + x^16 +...

%e where

%e A(x) = 1 + x + x^(2-1)*(1+x) + x^3 + x^(4-2)*(1+x)^2 + x^5 + x^(6-1)*(1+x) + x^7 + x^(8-3)*(1+x)^3 + x^9 + x^(10-1)*(1+x) + x^11 + x^(12-2)*(1+x)^2 + x^13 + x^(14-1)*(1+x) + x^(16-4)*(1+x)^4 +...

%e GENERATED AS A LIMIT FROM A227277.

%e Take A227277, ignore the initial 2 terms, then format as a triangle having 2^n terms in row n>=0; the resulting triangle begins:

%e 1;

%e 2, 1;

%e 3, 2, 2, 1;

%e 4, 4, 3, 1, 3, 2, 2, 1;

%e 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1;

%e 6, 11, 12, 6, 4, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1;

%e 7, 16, 22, 16, 9, 3, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 6, 11, 12, 6, 4, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1; ...

%e Note that the rows, when read in reverse, tend to this sequence as a limit.

%o (PARI) {a(n)=polcoeff(1+sum(k=1,n+#binary(n),x^(k-valuation(k,2))*(1+x)^valuation(k,2)+x*O(x^n)),n)}

%o for(n=0,128,print1(a(n),", "))

%Y Cf. A227277, A227318, A007814.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 04 2013