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URL: https://oeis.org/A227287

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A227287
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.
3
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, 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, 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
OFFSET
0,2
COMMENTS
The g.f. of related sequence A227277 is: Sum_{n>=0} x^n*(1+x)^A007814(n).
LINKS
FORMULA
Limit of rows, when read in reverse, of A227277 when formatted into a triangle having 2^n terms in row n>=0.
EXAMPLE
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 +...
where
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 +...
GENERATED AS A LIMIT FROM A227277.
Take A227277, ignore the initial 2 terms, then format as a triangle having 2^n terms in row n>=0; the resulting triangle begins:
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, 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, 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; ...
Note that the rows, when read in reverse, tend to this sequence as a limit.
PROG
(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)}
for(n=0, 128, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 04 2013
STATUS
approved