The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A205478 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A205478

G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n) ).
(history; published version)

#14 by Jon E. Schoenfield at Mon Oct 08 21:08:58 EDT 2018

STATUS

editing

approved

#13 by Jon E. Schoenfield at Mon Oct 08 21:08:56 EDT 2018

COMMENTS

Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^n) ) does not yield an integer sequenceseries.

STATUS

approved

editing

#12 by Jon E. Schoenfield at Sun Oct 07 20:44:26 EDT 2018

STATUS

proposed

approved

#11 by Jon E. Schoenfield at Sun Oct 07 20:23:06 EDT 2018

STATUS

editing

proposed

#10 by Jon E. Schoenfield at Sun Oct 07 20:23:03 EDT 2018

COMMENTS

Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^n) ) does not yield an integer seriessequence.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + ...

log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)/2 + x^3*(1+x^3)*(1+3*x^3)/3 + x^4*(1+x^4)*(1+2*x^4)*(1+4*x^4)/4 + x^5*(1+x^5)*(1+5*x^5)/5 + x^6*(1+x^6)*(1+2*x^6)*(1+3*x^6)*(1+6*x^6)/6 + ...

log(A(x)) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 + x^7/7 + 15*x^8/8 + 10*x^9/9 + 13*x^10/10 + ... + A205479(n)*x^n/n + ...

STATUS

approved

editing

#9 by Bruno Berselli at Wed Dec 23 02:55:31 EST 2015

STATUS

reviewed

approved

#8 by Joerg Arndt at Wed Dec 23 02:51:09 EST 2015

STATUS

proposed

reviewed

#7 by Jean-François Alcover at Wed Dec 23 02:29:05 EST 2015

STATUS

editing

proposed

#6 by Jean-François Alcover at Wed Dec 23 02:29:00 EST 2015

NAME

G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n) ).

COMMENTS

Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^n) ) does not yield an integer series.

FORMULA

Logarithmic derivative yields A205479.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 +...

MATHEMATICA

max = 50; s = Exp[Sum[(x^n/n)*Product[1+d*x^n, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 23 2015 *)

PROG

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

CROSSREFS

Cf. A205479, A198296, A205476, A205480, A205482, A205484, A205486, A205488, A205490.

STATUS

approved

editing

#5 by Russ Cox at Fri Mar 30 18:37:34 EDT 2012

AUTHOR

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jan 27 2012

Discussion

Fri Mar 30

18:37

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