_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jul 19 2011

proposed

approved

editing

proposed

0,1,3

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proposed

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.

G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 63*x^5 + 866*x^6 + 18444*x^7 +...

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) +...

(PARI) {a(n)=local(A=[1], F=x, G=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A);

Cf. A193192, A193193, A193194.

proposed

editing

editing

proposed

allocated for Paul D. Hanna

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.

1, 1, 2, 8, 63, 866, 18444, 559083, 22773527, 1197061138, 78782852673, 6341384941543, 612605031308910, 69931195961966196, 9310803519433216321, 1429869869684956113511, 250857267705012344767575, 49858270430813771746874366, 11143529422156562195864991584

0,3

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^2) +...

Explicitly,

A(A(x)) = x + 2*x^2 + 6*x^3 + 27*x^4 + 196*x^5 + 2379*x^6 + 46224*x^7 +...

(PARI) {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])}

Cf. A193192, A193193, A193194.

allocated

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 19 2011

approved

editing

allocated for Paul D. Hanna

allocated

approved