OFFSET
0,3
COMMENTS
Conjecture: this sequence consists entirely of integers.
Note that the following g.f. does NOT yield an integer series:
exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^k * x^k) * x^n/n ).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..120
EXAMPLE
G.f. A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 23*x^5 + 88*x^6 + ...
The logarithm of g.f. A(x) begins:
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 66*x^5/5 + 357*x^6/6 + 1891*x^7/7 + ... + A181071(n)*x^n/n + ...
and equals the series:
log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
+ (1 + 3^2*x + 3^3*x^2 + x^3)*x^3/3
+ (1 + 4^2*x + 6^3*x^2 + 4^4*x^3 + x^4)*x^4/4
+ (1 + 5^2*x + 10^3*x^2 + 10^4*x^3 + 5^5*x^4 + x^5)*x^5/5
+ (1 + 6^2*x + 15^3*x^2 + 20^4*x^3 + 15^5*x^4 + 6^6*x^5 + x^6)*x^6/6 + ...
MATHEMATICA
With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(k+1)*x^(n+k)/n, {k, 0, m+2}], {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)^(k+1)*x^k)*x^m/m)+x*O(x^n)), n)}
(Magma)
m:=30;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n, k)^(k+1)*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
(Sage)
m=30;
def A181070_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( exp( sum( sum( binomial(n, k)^(k+1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181070_list(m) # G. C. Greubel, Apr 05 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 02 2010
STATUS
approved