OFFSET
0,3
COMMENTS
In general, if A^n = BINOMIAL(A^(n-1)), then for all integer m>0 there exists an integer sequence B such that B^d = BINOMIAL(A^m) where d=gcd(m+1,n). Also, coefficients of A(k*x)^n = k-th binomial transform of coefficients in A(k*x)^(n-1) for all k>0.
In general, if g.f. satisfies A(x)^(k+1) = A(x/(1-x))^k / (1-x), k>0, then a(n) ~ (n-1)! / (k*(k+1) * (log((k+1)/k))^(n+1)). - Vaclav Kotesovec, Nov 19 2014
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..280
FORMULA
G.f. satisfies: A(x)^6 = A(x/(1-x))^5/(1-x).
a(n) ~ (n-1)! / (30 * (log(6/5))^(n+1)). - Vaclav Kotesovec, Nov 19 2014
O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = Sum_{k = 1..n} k!*Stirling2(n,k)*5^(k-1) = 1/5*A094418(n) for n >= 1. - Peter Bala, May 26 2015
EXAMPLE
MATHEMATICA
nmax = 16; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^6 - A[x/(1 - x)]^5/(1 - x) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)
With[{m=40}, CoefficientList[Series[Exp[Sum[Sum[5^(j-1)*j!* StirlingS2[k, j], {j, k}]*x^k/k, {k, m+1}]], {x, 0, m}], x]] (* G. C. Greubel, Jun 08 2023 *)
PROG
(PARI) {a(n)=local(A); if(n<1, 0, A=1+x+x*O(x^n); for(k=1, n, B=subst(A^5, x, x/(1-x))/(1-x)+x*O(x^n); A=A-A^6+B); polcoeff(A, n, x))}
(Magma)
m:=40;
f:= func< n, x | Exp((&+[(&+[5^(j-1)*Factorial(j)*StirlingSecond(k, j) *x^k/k: j in [1..k]]): k in [1..n+2]])) >;
R<x>:=PowerSeriesRing(Rationals(), m+1); // A090358
Coefficients(R!( f(m, x) )); // G. C. Greubel, Jun 08 2023
(SageMath)
m=50
def f(n, x): return exp(sum(sum( 5^(j-1)*factorial(j)* stirling_number2(k, j)*x^k/k for j in range(1, k+1)) for k in range(1, n+2)))
def A090358_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(m, x) ).list()
A090358_list(m-5) # G. C. Greubel, Jun 08 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Nov 26 2003
STATUS
approved