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

1, 1, 2, 6, 21, 78, 308, 1264, 5332, 22994, 100896, 449004, 2021712, 9193509, 42161222, 194768936, 905522052, 4233712140, 19893553120, 93894821200, 444952447944, 2116220266360, 10098086643002, 48330679370584, 231954451580616, 1116046254269592, 5382402925982248

Offset: 0

Author

Paul D. Hanna, Dec 02 2022

Keywords

• nonn

Comments

Is this sequence the same as A287211?

Examples

			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 + ...
where A(x) = 1 + x*(2*A(x)^3 + A(x^3))/3.
RELATED SERIES.
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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A287211, A000625, A375439.

Programs

Maple

A357538 := proc(n)
    option remember ;
    if n < 0 then
        0;
    elif n <= 1 then
        1;
    else
        a := 0 ;
        for j from 0 to n-1 do
            a := a + procname(n-1-j)*add(procname(i)*procname(j-i),i=0..j)
        end do:
        a := 2*a/3 ;
        if modp(n-1,3) = 0 then
            a := a+procname((n-1)/3)/3 ;
        end if;
        a ;
    end if ;
end proc:
seq(A357538(n),n=0..20) ; # R. J. Mathar, Dec 19 2022

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)}
for(n=0,30,print1(a(n),", "))

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
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = 1 + x*(2*A(x)^3 + A(x^3))/3.

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)