A363557 - OEIS

%I #9 Jul 20 2023 10:07:15

%S 1,1,1,2,4,9,20,47,112,273,677,1702,4330,11128,28847,75341,198066,

%T 523713,1391869,3716098,9962252,26806275,72372721,195994320,532266707,

%U 1449216287,3955193019,10818202369,29650108510,81417795070,223964216673,617097850848,1702943168118

%N Expansion of g.f. A(x) satisfying 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^n / Product_{k=1..n+1} (1 - x^k).

%C Related identities:

%C (1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * B(x)^n / Product_{k=1..n+1} (1 - x^k*B(x)), where B(x) = 1/(1-x).

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

%H Paul D. Hanna, <a href="/A363557/b363557.txt">Table of n, a(n) for n = 0..500</a>

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

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

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

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 47*x^7 + 112*x^8 + 273*x^9 + 677*x^10 + 1702*x^11 + 4330*x^12 + ...

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

%o A[#A] = polcoeff( sum(m=0,2*sqrtint(#A), (-1)^m * (x)^(m*(m-1)/2) * Ser(A)^m / prod(k=1,m+1, (1 - x^k +x*O(x^#A) ) )),#A-1);); A[n+1]}

%o for(n=0,32,print1(a(n),", "))

%Y Cf. A082395, A363555.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jul 11 2023