The On-Line Encyclopedia of Integer Sequences (OEIS)

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G.f.: exp( Sum_{n>=1} A001511(n)*2^A001511(n)*x^n/n ) where A001511(n) equals the 2-adic valuation of 2n.
(history; published version)

#10 by Russ Cox at Fri Mar 30 18:37:23 EDT 2012

AUTHOR

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Dec 19 2010

Discussion

Fri Mar 30

18:37

OEIS Server: https://oeis.org/edit/global/213

#9 by T. D. Noe at Mon Dec 20 13:37:57 EST 2010

STATUS

reviewed

approved

#8 by Joerg Arndt at Mon Dec 20 03:38:57 EST 2010

STATUS

proposed

reviewed

#7 by Paul D. Hanna at Sun Dec 19 21:09:13 EST 2010

CROSSREFS

Cf. A183037, A183038, A001511, A000123.

#6 by Paul D. Hanna at Sun Dec 19 21:06:50 EST 2010

FORMULA

G.f. satisfies: A(x) = (1+-x^2)/(1-x) ^2 * A(x^2)^2/A(x^4).

#5 by Paul D. Hanna at Sun Dec 19 20:54:58 EST 2010

COMMENTS

Note: 2n/2^A001511(n) is odd for n>=1, so that A001511(n) is logarithmic in nature.

2^A001511(n) exactly divides 2n.

FORMULA

G.f. satisfies: A(x) = (1+x)/(1-x) * A(x^2)^2/A(x^4).

#4 by Paul D. Hanna at Sun Dec 19 18:02:33 EST 2010

NAME

G.f.: exp( Sum_{n>=1} A001511(n)*2^A001511(n)*x^n/n ) where A001511(n) equals the 2-adic valuation of 2n.

COMMENTS

Compare to B(x), the g.f. of the binary partitions (A000123):

Note: 2n/2^A001511(n) is odd for n>=1, so that A001511(n) is logarithmic in nature.

EXAMPLE

G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 24*x^4 + 38*x^5 + 74*x^6 +...

PROG

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

CROSSREFS

Cf. A183037, A001511, A000123.

#3 by Paul D. Hanna at Sun Dec 19 17:50:00 EST 2010

NAME

allocated for Paul D. Hanna

G.f.: exp( Sum_{n>=1} A001511(n)*2^A001511(n)*x^n/n ) where A001511(n) equals the 2-adic valuation of 2n.

DATA

1, 2, 6, 10, 24, 38, 74, 110, 200, 290, 486, 682, 1096, 1510, 2314, 3118, 4650, 6182, 8946, 11710, 16616, 21522, 29886, 38250, 52328, 66406, 89394, 112382, 149496, 186610, 245086, 303562, 394814, 486066, 625686, 765306, 977112, 1188918, 1504954

OFFSET

0,2

COMMENTS

Compare to B(x), the g.f. of the binary partitions (A000123):

B(x) = exp( Sum_{n>=1} 2^A001511(n)*x^n/n ) = (1-x)^(-1)*Product_{n>=0} 1/(1 - x^(2^n)).

EXAMPLE

G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 24*x^4 + 38*x^5 + 74*x^6 +...

log(A(x)) = 2*x + 8*x^2/2 + 2*x^3/3 + 24*x^4/4 + 2*x^5/5 + 8*x^6/6 + 2*x^7/7 + 64*x^8/8 + 2*x^9/9 + 8*x^10/10 +...+ A183037(n)*x^n/n +...

PROG

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

CROSSREFS

Cf. A183037, A001511, A000123.

KEYWORD

allocated

nonn

AUTHOR

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 19 2010

STATUS

approved

proposed

#2 by Paul D. Hanna at Sun Dec 19 17:45:01 EST 2010

KEYWORD

allocating

allocated

#1 by Paul D. Hanna at Sun Dec 19 17:45:01 EST 2010

NAME

allocated for Paul D. Hanna

KEYWORD

allocating

STATUS

approved