cp's OEIS Frontend

A214693 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

%I A214693 #8 Jan 15 2025 11:24:39
%S A214693 1,1,4,34,338,3691,42623,510949,6289912,78972928,1006665781,
%T A214693 12985611054,169115724583,2219614920740,29318819296959,
%U A214693 389331204757856,5192978617937181,69522908878900079,933674035184058960,12571898958515379108,169651868248129552194
%N A214693 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).
%C A214693 Compare the g.f. to the identity:
%C A214693 G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
%C A214693 which holds for all power series G(x) such that G(0)=1.
%F A214693 G.f. satisfies: 1+x = A(y) where y = x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9, which is the g.f. of row 3 in triangle A214690.
%F A214693 G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+6)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).
%e A214693 G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 338*x^4 + 3691*x^5 + 42623*x^6 +...
%e A214693 The g.f. satisfies:
%e A214693 x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^3-1)/A(x)^16 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^27 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^40 +
%e A214693 (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^55 +...
%o A214693 (PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9 +x^2*O(x^n)), n))}
%o A214693 (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(6*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
%o A214693 for(n=0, 25, print1(a(n), ", "))
%Y A214693 Cf. A214690, A214692, A214694, A214695, A181997 (variant).
%K A214693 nonn
%O A214693 0,3
%A A214693 _Paul D. Hanna_, Jul 26 2012