A395121 - OEIS

%I #10 Jun 09 2026 10:12:18

%S 1,1,3,16,137,1746,30787,708289,20393697,713704924,29707484531,

%T 1446005259579,81182236620817,5197625956902718,375890959014970323,

%U 30458434507868797231,2745996328319300018913,273765648582428330478776,30018774941547810310868707,3602830336301261670977620531

%N a(n) = A395120(n+1)/(n+1) for n >= 0.

%H Paul D. Hanna, <a href="/A395121/b395121.txt">Table of n, a(n) for n = 0..100</a>

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1746*x^5/5! + 30787*x^6/6! + 708289*x^7/7! + 20393697*x^8/8! + ...

%e Let B(x) = x*A(x) then

%e A(x) = 1 + B(x) + x*B(B(x))/2! + x^2*B(B(B(x)))/3! + x^3*B(B(B(B(x))))/4! + ... + x^(n-1)*B^n(x)/n! + ....

%e where

%e B(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 685*x^5/5! + 10476*x^6/6! + ... + A395120(n)*x^n/n! + ...

%e satisfies B(x) = Sum_{n>=0} B^n(x) * x^n/n!.

%o (PARI) \\ Routine to calculate the n-th iteration

%o {IT(n,F) = my(G=x); for(i=1,n, G = subst(F,x,G)); G}

%o \\ Get e.g.f. from iterations of B(x) (e.g.f. of A395120)

%o {a(n) = my(B=x); for(k=1,n, B = truncate(B) + x*O(x^k);

%o B = sum(m=0,k, x^m/m! * IT(m,B) ) ); polcoef(EGF=B/x,n)}

%o {upto(n) = a(n); Vec(serlaplace(EGF))}

%o upto(20)

%Y Cf. A395120.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 10 2026