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A357538 a(n) = coefficient of x^n in A(x) such that A(x) = 1 + x*(2*A(x)^3 + A(x^3))/3.

%I A357538 #23 Aug 22 2024 02:06:53
%S A357538 1,1,2,6,21,78,308,1264,5332,22994,100896,449004,2021712,9193509,
%T A357538 42161222,194768936,905522052,4233712140,19893553120,93894821200,
%U A357538 444952447944,2116220266360,10098086643002,48330679370584,231954451580616,1116046254269592,5382402925982248
%N A357538 a(n) = coefficient of x^n in A(x) such that A(x) = 1 + x*(2*A(x)^3 + A(x^3))/3.
%C A357538 Is this sequence the same as A287211?
%H A357538 Paul D. Hanna, <a href="/A357538/b357538.txt">Table of n, a(n) for n = 0..1000</a>
%F A357538 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = 1 + x*(2*A(x)^3 + A(x^3))/3.
%F A357538 a(0) = a(1) = 1; a(n+1) = a(n/3)/3 + 2*(Sum_{j=0..n} a(n-j)*(Sum_{i=0..j} a(i)*a(j-i)))/3 for n >= 1, where a(k) = 0 if k not an integer (see formula and comment by _Emeric Deutsch_ in A000625). (corrected by _R. J. Mathar_, Dec 19 2022)
%e A357538 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 78*x^5 + 308*x^6 + 1264*x^7 + 5332*x^8 + 22994*x^9 + 100896*x^10 + ...
%e A357538 where A(x) = 1 + x*(2*A(x)^3 + A(x^3))/3.
%e A357538 RELATED SERIES.
%e A357538 A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 117*x^4 + 462*x^5 + 1895*x^6 + 7998*x^7 + 34491*x^8 + 151341*x^9 + 673506*x^10 + ...
%p A357538 A357538 := proc(n)
%p A357538     option remember ;
%p A357538     if n < 0 then
%p A357538         0;
%p A357538     elif n <= 1 then
%p A357538         1;
%p A357538     else
%p A357538         a := 0 ;
%p A357538         for j from 0 to n-1 do
%p A357538             a := a + procname(n-1-j)*add(procname(i)*procname(j-i),i=0..j)
%p A357538         end do:
%p A357538         a := 2*a/3 ;
%p A357538         if modp(n-1,3) = 0 then
%p A357538             a := a+procname((n-1)/3)/3 ;
%p A357538         end if;
%p A357538         a ;
%p A357538     end if ;
%p A357538 end proc:
%p A357538 seq(A357538(n),n=0..20) ; # _R. J. Mathar_, Dec 19 2022
%o A357538 (PARI) {a(n) = my(A=1); for(i=1,n, A = 1 + x*(2*A^3 + subst(A,x,x^3))/3 +x*O(x^n)); polcoeff(A,n)}
%o A357538 for(n=0,30,print1(a(n),", "))
%o A357538 (PARI) {a(n) = if(n, my(A=vector(n+1)); A[1]=1; A[2]=1; for(k=1, n-1, A[k+2] = sum(j=1, k, 2*j*A[j+1]*(sum(i=0, k-j, A[i+1]*A[k-j-i+1])))/k + (1/3)*if(k%3, 0, A[k/3+1])); A[n+1], 1)} \\ after _Jianing Song_ in A000625
%o A357538 for(n=0,30,print1(a(n),", "))
%Y A357538 Cf. A287211, A000625, A375439.
%K A357538 nonn
%O A357538 0,3
%A A357538 _Paul D. Hanna_, Dec 02 2022