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} C(n+k-1,k)^k * x^k] * x^n/n ).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..75
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 231*x^5 + 5405*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 71*x^4/4 + 1026*x^5/5 + 30912*x^6/6 +...+ A181075(n)*x^n/n +...
which equals the series:
log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
+ (1 + 2^2*x + 3^3*x^2 + 4^4*x^3 + 5^5*x^4 + 6^6*x^5 + ...)*x^2/2
+ (1 + 3^2*x + 6^3*x^2 + 10^4*x^3 + 15^5*x^4 + 21^6*x^5 + ...)*x^3/3
+ (1 + 4^2*x + 10^3*x^2 + 20^4*x^3 + 35^5*x^4 + 56^6*x^5 + ...)*x^4/4
+ (1 + 5^2*x + 15^3*x^2 + 35^4*x^3 + 70^5*x^4 + 126^6*x^5 + ...)*x^5/5
+ (1 + 6^2*x + 21^3*x^2 + 56^4*x^3 + 126^5*x^4 + 252^6*x^5 + ...)*x^6/6
+ (1 + 7^2*x + 28^3*x^2 + 84^4*x^3 + 210^5*x^4 + 462^6*x^5 + ...)*x^7/7 + ...
MATHEMATICA
With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1, 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, n, binomial(m+k-1, 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-1, k)^(k+1)*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
(SageMath)
m=30;
def A181066_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( exp( sum( sum( binomial(n+k-1, k)^(k+1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181066_list(m) # G. C. Greubel, Apr 05 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 02 2010
STATUS
approved