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%I #13 Oct 24 2020 04:42:40
%S 1,1,2,23,383,8456,228657,7268077,264627570,10842464809,493454895161,
%T 24695915073150,1348165575801780,79740451015249487,
%U 5080946666286668723,347046897108675507548,25300042941361404342404,1960975025091063811051791,161045636717926154800016147,13970534533583992498224467231
%N G.f.: A(x) satisfies 1 = Sum_{n>=0} Product_{k=n..2*n-1} ( (1+x)^k - A(x) ).
%C Compare the g.f. A(x) to functions B(x) and C(x) defined by
%C A338181: 1 = Sum_{n>=0} Product_{k=2*n..3*n-1} ((1+x)^k - B(x)) ;
%C A338183: 1 = Sum_{n>=0} Product_{k=3*n..4*n-1} ((1+x)^k - C(x)).
%H Paul D. Hanna, <a href="/A338178/b338178.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} Product_{k=n..2*n-1} ( (1+x)^k - A(x) ).
%F (2) 1 = Sum_{n>=0} (1+x)^(n*(3*n-1)/2) / Product_{k=n..2*n} (1 + (1+x)^k*A(x)).
%F a(n) ~ c * d^n * n! / sqrt(n), where d = 4.6634423082484267335372948079179... and c = 0.10798828318774... - _Vaclav Kotesovec_, Oct 24 2020
%e G.f.: A(x) = 1 + x + 2*x^2 + 23*x^3 + 383*x^4 + 8456*x^5 + 228657*x^6 + 7268077*x^7 + 264627570*x^8 + 10842464809*x^9 + 493454895161*x^10 + ...
%e Let A = A(x), then g.f. A(x) satisfies
%e 1 = 1 + ((1+x) - A) + ((1+x)^2 - A)*((1+x)^3 - A) + ((1+x)^3 - A)*((1+x)^4 - A)*((1+x)^5 - A) + ((1+x)^4 - A)*((1+x)^5 - A)*((1+x)^6 - A)*((1+x)^7 - A) + ((1+x)^5 - A)*((1+x)^6 - A)*((1+x)^7 - A)*((1+x)^8 - A)*((1+x)^9 - A) + ... + Product_{k=n..2*n-1} ((1+x)^k - A(x)) + ...
%e also
%e 1 = 1/(1 + A) + (1+x)/((1 + (1+x)*A)*(1 + (1+x)^2*A)) + (1+x)^5/((1 + (1+x)^2*A)*(1 + (1+x)^3*A)*(1 + (1+x)^4*A)) + (1+x)^12/((1 + (1+x)^3*A)*(1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A)) + (1+x)^22/((1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A)*(1 + (1+x)^7*A)*(1 + (1+x)^8*A)) + ... + (1+x)^(n*(3*n-1)/2)/( Product_{k=n..2*n} (1 + (1+x)^(k*A(x)) ) + ...
%o (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, prod(k=m,2*m-1,(1+x)^k - Ser(A)) ) )[#A] ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A338181, A338183.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 14 2020
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