The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A205482 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A205482

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

#13 by Jon E. Schoenfield at Mon Oct 08 21:07:46 EDT 2018

STATUS

editing

approved

#12 by Jon E. Schoenfield at Mon Oct 08 21:07:44 EDT 2018

COMMENTS

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

STATUS

approved

editing

#11 by Jon E. Schoenfield at Sun Oct 07 20:44:02 EDT 2018

STATUS

proposed

approved

#10 by Jon E. Schoenfield at Sun Oct 07 20:24:35 EDT 2018

STATUS

editing

proposed

#9 by Jon E. Schoenfield at Sun Oct 07 20:24:33 EDT 2018

COMMENTS

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

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 + 15*x^6 + 15*x^7 + ...

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

log(A(x)) = x + 3*x^2/2 + x^3/3 + 11*x^4/4 + x^5/5 + 45*x^6/6 + x^7/7 + 59*x^8/8 + 109*x^9/9 + 53*x^10/10 + ... + A205483(n)*x^n/n + ...

STATUS

approved

editing

#8 by Bruno Berselli at Wed Dec 23 03:22:53 EST 2015

STATUS

proposed

approved

#7 by Jean-François Alcover at Wed Dec 23 03:07:54 EST 2015

STATUS

editing

proposed

#6 by Jean-François Alcover at Wed Dec 23 03:07:46 EST 2015

NAME

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

COMMENTS

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

FORMULA

Logarithmic derivative yields A205483.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 + 15*x^6 + 15*x^7 +...

MATHEMATICA

max = 40; s = Exp[Sum[(x^n/n)*Product[(1 + d*x^n)^d, {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, d*log(1+d*x^m+x*O(x^n)))))), n)}

CROSSREFS

Cf. A205483 (log), A205476, A205478, A205480, 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

#4 by N. J. A. Sloane at Sat Jan 28 10:31:24 EST 2012

STATUS

proposed

approved