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.

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

Paul D. Hanna, Table of n, a(n) for n = 0..10000

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

Cf. A227277, A227318, A007814.

Sequence in context: A080045 A191384 A191305 * A289236 A280172 A337942

Adjacent sequences: A227284 A227285 A227286 * A227288 A227289 A227290

Keyword

nonn

Author

Paul D. Hanna, Jul 04 2013

Status

approved