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Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} (-x)^n * A(x)^n / Product_{k=1..n+1} (1 + (-x)^k).
1

%I #15 Jun 25 2023 19:31:28

%S 1,1,2,3,5,9,18,37,78,168,367,809,1800,4040,9136,20798,47627,109630,

%T 253506,588602,1371688,3207309,7522302,17691875,41716878,98600132,

%U 233557375,554358994,1318282031,3140435548,7493519010,17908158678,42859216345,102713690586,246472881089

%N Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} (-x)^n * A(x)^n / Product_{k=1..n+1} (1 + (-x)^k).

%C A related identity: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^n / Product_{k=1..n+1} (1 + (-x)^k*A(x) ), which holds formally for all A(x).

%H Paul D. Hanna, <a href="/A363307/b363307.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.

%F (1) 1 = Sum_{n>=0} (-x)^n * A(x)^n / Product_{k=1..n+1} (1 + (-x)^k).

%F (2) 1 = Sum_{n>=0} x^n / Product_{k=1..n+1} (1 - (-x)^k*A(x)).

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 18*x^6 + 37*x^7 + 78*x^8 + 168*x^9 + 367*x^10 + 809*x^11 + 1800*x^12 + ...

%e where

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

%e also,

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

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff(-1 + sum(n=0,#A, (-x)^n*Ser(A)^n / prod(k=1,n+1,(1 + (-x)^k ) ) ),#A););A[n+1]}

%o for(n=0,35, print1(a(n),", "))

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff(-1 + sum(n=0,#A, x^n / prod(k=1,n+1,(1 - (-x)^k*Ser(A)) ) ),#A););A[n+1]}

%o for(n=0,35, print1(a(n),", "))

%Y Cf. A352018.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 25 2023