A227277 - OEIS

%I #32 Jul 06 2013 14:59:32

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

%T 11,12,6,4,2,2,1,4,4,3,1,3,2,2,1,5,7,6,2,3,2,2,1,4,4,3,1,3,2,2,1,7,16,

%U 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

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

%H Paul D. Hanna, <a href="/A227277/b227277.txt">Table of n, a(n) for n = 0..10000</a>

%F a(4*A004772(n)) = 1 where A004772 lists numbers that are not congruent to 1 mod 4.

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

%e where

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

%e AS AN IRREGULAR TRIANGLE.

%e When formatted into a triangle with rows of 2^n terms, we see patterns more clearly.

%e Sequence starts with 1, 1, followed by terms in the following triangle:

%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 in which row n+1 can be generated from row n by:

%e * T(n+1,k) = T(n,k) + binomial(n,k)  for 0<= k <=2^n, and

%e * T(n+1,2^n+k) = T(n,k)  for 0<= k <=2^n.

%e Note that the rows, when read in reverse, tend to have a limit (A227287):

%e [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, 2, 2, 3, 1, 3, 4, 4, 1, ...];

%e where the g.f. of the resulting sequence is:

%e Sum_{n>=0} x^(n - b(n)) * (1+x)^b(n), where b(n) = A007814(n).

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

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

%Y Cf. A227287, A227318, A007814.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jul 04 2013