cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A341378 G.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (n+1)*x^n*(1 + x*A(x)^n)^n and Q(x) = Sum_{n>=0} x^n*(1 + x*A(x)^n)^n.

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%I A341378 #7 Feb 20 2021 17:22:19
%S A341378 1,1,2,4,13,49,198,847,3796,17661,84754,417556,2104652,10825545,
%T A341378 56714218,302185765,1635727560,8987357912,50090647729,283058794945,
%U A341378 1621221821820,9409181197452,55327487031279,329591827614844,1989063796779401
%N A341378 G.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (n+1)*x^n*(1 + x*A(x)^n)^n and Q(x) = Sum_{n>=0} x^n*(1 + x*A(x)^n)^n.
%F A341378 G.f. A(x) satisfies:
%F A341378 (1) A(x) = P(x)/Q(x) where
%F A341378 P(x) = Sum_{n>=0} (n+1) * x^n * (1 + x*A(x)^n)^n and
%F A341378 Q(x) = Sum_{n>=0} x^n * (1 + x*A(x)^n)^n.
%F A341378 (2) A(x) = P(x)/Q(x) where
%F A341378 P(x) = Sum_{n>=0} (n+1) * x^n * (1 + x*A(x)^n)^n and
%F A341378 Q(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2) / (1 - x*A^n)^(n+1).
%e A341378 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 13*x^4 + 49*x^5 + 198*x^6 + 847*x^7 + 3796*x^8 + 17661*x^9 + 84754*x^10 + 417556*x^11 + 2104652*x^12 + ...
%e A341378 such that A(x) = P(x)/Q(x) where
%e A341378 P(x) = 1 + 2*x*(1 + x*A(x)) + 3*x^2*(1 + x*A(x)^2)^2 + 4*x^3*(1 + x*A(x)^3)^3 + 5*x^4*(1 + x*A(x)^4)^4 + 6*x^5*(1 + x*A(x)^5)^5 + ...
%e A341378 Q(x) = 1 + x*(1 + x*A(x)) + x^2*(1 + x*A(x)^2)^2 + x^3*(1 + x*A(x)^3)^3 + x^4*(1 + x*A(x)^4)^4 + x^5*(1 + x*A(x)^5)^5 + ...
%e A341378 also
%e A341378 Q(x) = 1/(1 - x) + x^2*A(x)/(1 - x*A(x))^2 + x^4*A(x)^4/(1 - x*A(x)^2)^3 + x^6*A(x)^9/(1 - x*A(x)^3)^4 + x^8*A(x)^16/(1 - x*A(x)^4)^5 + ...
%e A341378 explicitly,
%e A341378 P(x) = 1 + 2*x + 5*x^2 + 12*x^3 + 36*x^4 + 124*x^5 + 471*x^6 + 1918*x^7 + 8246*x^8 + 37024*x^9 + 172266*x^10 + 825870*x^11 + 4062450*x^12 + ...
%e A341378 Q(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 35*x^5 + 125*x^6 + 484*x^7 + 1994*x^8 + 8625*x^9 + 38804*x^10 + 180360*x^11 + 861833*x^12 + ...
%o A341378 (PARI) {a(n) = my(A=1 +x*O(x^n)); for(i=1,n,
%o A341378 A = sum(m=0,n, (m+1) * x^m *(1 + x*A^m +x*O(x^n))^m ) / sum(m=0,n, x^m *(1 + x*A^m +x*O(x^n))^m )   );
%o A341378 polcoeff(H=A,n)}
%o A341378 for(n=0,20,print1(a(n),", "))
%Y A341378 Cf. A337851.
%K A341378 nonn
%O A341378 0,3
%A A341378 _Paul D. Hanna_, Feb 19 2021