A193193
G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n / (1-x)^(n^2), where g.f. A(x) = Sum_{n>=1} a(n)*x^n.
Original entry on oeis.org
1, 1, 3, 22, 334, 8831, 359836, 20845201, 1625007715, 163854289212, 20739421240200, 3218400384155498, 600776969761195428, 132793055529329858607, 34298178516935957467888, 10235014757932193318825335
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 22*x^4 + 334*x^5 + 8831*x^6 +...
where
A(A(x)) = x/(1-x) + x^2/(1-x)^4 + 3*x^3/(1-x)^9 + 22*x^4/(1-x)^16 + 334*x^5/(1-x)^25 + 8831*x^6/(1-x)^36 +...+ a(n)*x^n/(1-x)^(n^2) +...
Explicitly,
A(A(x)) = x + 2*x^2 + 8*x^3 + 60*x^4 + 842*x^5 + 20704*x^6 + 805796*x^7 +...
-
{a(n)=local(A=[1],F=x,G=x);for(i=1,n,A=concat(A,0);F=x*Ser(A);
G=sum(m=1,#A-1,A[m]*x^m/(1-x+x*O(x^#A))^(m^2));
A[#A]=Vec(G)[#A]-Vec(subst(F,x,F))[#A]);if(n<1,0,A[n])}
A193194
G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n * (1+x)^(n*(n+1)/2), where g.f. A(x) = Sum_{n>=1} a(n)*x^n.
Original entry on oeis.org
1, 1, 1, 3, 20, 262, 5367, 158413, 6318384, 326575077, 21199737973, 1687053244236, 161418184139781, 18276066372054109, 2416167457088427950, 368773198369779785338, 64348161941454274119082, 12728047101293068225626576, 2832615019902477894227329544
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 20*x^5 + 262*x^6 + 5367*x^7 +...
where
A(A(x)) = x*(1+x) + x^2*(1+x)^3 + x^3*(1+x)^6 + 3*x^4*(1+x)^10 + 20*x^5*(1+x)^15 + 262*x^6*(1+x)^21 +...+ a(n)*x^n*(1+x)^(n*(n+1)/2) +...
Explicitly,
A(A(x)) = x + 2*x^2 + 4*x^3 + 12*x^4 + 66*x^5 + 717*x^6 + 13344*x^7 +...
-
{a(n)=local(A=[1],F=x,G=x);for(i=1,n,A=concat(A,0);F=x*Ser(A);
G=sum(m=1,#A-1,A[m]*x^m*(1+x+x*O(x^#A))^(m*(m+1)/2));
A[#A]=Vec(G)[#A]-Vec(subst(F,x,F))[#A]);if(n<1,0,A[n])}
A193192
G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n * (1+x)^(n^2), where g.f. A(x) = Sum_{n>=1} a(n)*x^n.
Original entry on oeis.org
1, 1, 2, 13, 184, 4725, 188596, 10765407, 829780846, 82924007284, 10420182259194, 1607406366386249, 298555458341808338, 65711158773953092780, 16910051487116790543954, 5030141451818448773854244, 1712632076858599057432066794
Offset: 1
G.f.: A(x) = x + x^2 + 2*x^3 + 13*x^4 + 184*x^5 + 4725*x^6 +...
where
A(A(x)) = x*(1+x) + x^2*(1+x)^4 + 2*x^3*(1+x)^9 + 13*x^4*(1+x)^16 + 184*x^5*(1+x)^25 + 4725*x^6*(1+x)^36 +...+ a(n)*x^n*(1+x)^(n^2) +...
Explicitly,
A(A(x)) = x + 2*x^2 + 6*x^3 + 37*x^4 + 468*x^5 + 11054*x^6 + 421428*x^7 +...
-
{a(n)=local(A=[1],F=x,G=x);for(i=1,n,A=concat(A,0);F=x*Ser(A);
G=sum(m=1,#A-1,A[m]*x^m*(1+x+x*O(x^#A))^(m^2));
A[#A]=Vec(G)[#A]-Vec(subst(F,x,F))[#A]);if(n<1,0,A[n])}
Showing 1-3 of 3 results.