The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A181084 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} binomial(n,k)^(n+k+1) * x^k] * x^n/n ). 3
1, 1, 2, 10, 92, 1367, 87090, 20385333, 6633475836, 4096297538926, 14834973644512627, 119919823546238898903, 1273371038284317852447990, 41086272137585936052959008420, 6982122140549374036504235218052104 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Conjecture: this sequence consists entirely of integers.
LINKS
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...
The logarithm of g.f. A(x) begins:
log(A(x)) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + 513657*x^6/6 + ... + A181085(n)*x^n/n + ...
and equals the series:
log(A(x)) = (1 + x)*x + (1 + 2^4*x + x^2)*x^2/2
+ (1 + 3^5*x + 3^6*x^2 + x^3)*x^3/3
+ (1 + 4^6*x + 6^7*x^2 + 4^8*x^3 + x^4)*x^4/4
+ (1 + 5^7*x + 10^8*x^2 + 10^9*x^3 + 5^10*x^4 + x^5)*x^5/5
+ (1 + 6^8*x + 15^9*x^2 + 20^10*x^3 + 15^11*x^4 + 6^12*x^5 + x^6)*x^6/6 + ...
MATHEMATICA
With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(n+k+1)*x^(n+k)/n, {k, 0, n}], {n, m+1}]], {x, 0, m}], x]] (* G. C. Greubel, Apr 05 2021 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^(m+k+1)*x^k)*x^m/m) + x*O(x^n)), n)}
(Magma)
m:=20;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n, k)^(n+k+1)*x^(n+k)/n : k in [0..n]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
(Sage)
m=20;
def A181084_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( exp( sum( sum( binomial(n, k)^(n+k+1)*x^(n+k)/n for k in (0..n) ) for n in (1..m+1)) ) ).list()
A181084_list(m) # G. C. Greubel, Apr 05 2021
CROSSREFS
Cf. A181085 (log), variants: A181080, A181082.
Sequence in context: A289020 A195415 A336271 * A063385 A293709 A063393
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 28 2010
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 15 01:02 EDT 2024. Contains 373402 sequences. (Running on oeis4.)