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])}
A193195
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, 2, 8, 63, 866, 18444, 559083, 22773527, 1197061138, 78782852673, 6341384941543, 612605031308910, 69931195961966196, 9310803519433216321, 1429869869684956113511, 250857267705012344767575, 49858270430813771746874366, 11143529422156562195864991584
Offset: 1
G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 63*x^5 + 866*x^6 + 18444*x^7 +...
where
A(A(x)) = x/(1-x) + x^2/(1-x)^3 + 2*x^3/(1-x)^6 + 8*x^4/(1-x)^10 + 63*x^5/(1-x)^15 + 866*x^6/(1-x)^21 +...+ a(n)*x^n/(1-x)^(n*(n+1)/2) +...
Explicitly,
A(A(x)) = x + 2*x^2 + 6*x^3 + 27*x^4 + 196*x^5 + 2379*x^6 + 46224*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])}
Showing 1-3 of 3 results.