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G.f. satisfies: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} (A(x)^k - x^k).
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%I #10 Aug 16 2024 18:55:23

%S 1,1,1,4,13,54,227,1019,4762,23012,114487,583893,3044598,16197880,

%T 87798933,484377698,2718044549,15507602279,89947655628,530420332747,

%U 3180793948207,19404122501300,120475202020535,761689102469013,4906578443997601,32221210135485056,215818333117100371

%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} (A(x)^k - x^k).

%e G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 13*x^4 + 54*x^5 + 227*x^6 + 1019*x^7 +...

%e where g.f. A = A(x) satisfies:

%e A(x) = 1 + x*(A-x) + x^2*(A-x)*(A^2-x^2) + x^3*(A-x)*(A^2-x^2)*(A^3-x^3) + x^4*(A-x)*(A^2-x^2)*(A^3-x^3)*(A^4-x^4) + x^5*(A-x)*(A^2-x^2)*(A^3-x^3)*(A^4-x^4)*(A^5-x^5) +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*prod(k=1,m,A^k-x^k +x*O(x^n))));polcoeff(A,n)}

%o for(n=0,40,print1(a(n),", "))

%K nonn

%O 0,4

%A _Paul D. Hanna_, Sep 15 2013