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a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^(n+1))^n * A(x)^(n+1).
3

%I #7 Dec 24 2022 11:17:07

%S 1,3,15,114,1086,10824,114382,1252002,14083275,161810358,1890774909,

%T 22401092826,268465408738,3248818848876,39643793276526,

%U 487251937616006,6026537732208078,74954027622814455,936840765257368687,11761260253206563461,148240496011414115676

%N a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^(n+1))^n * A(x)^(n+1).

%C Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (y - x^n)^(n-1), which holds formally for all y.

%H Paul D. Hanna, <a href="/A357794/b357794.txt">Table of n, a(n) for n = 0..300</a>

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

%F (1) 1 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^(n+1))^n * A(x)^(n+1).

%F (2) 1 = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)/2 * x^(n*(n-2)) / ((1 - x^(n-1))^n * A(x)^(n-1)).

%e G.f.: A(x) = 1 + 3*x + 15*x^2 + 114*x^3 + 1086*x^4 + 10824*x^5 + 114382*x^6 + 1252002*x^7 + 14083275*x^8 + 161810358*x^9 + 1890774909*x^10 + ...

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

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

%o for(n=0, 30, print1(a(n), ", "))

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

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

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A357158, A357795, A357796.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 22 2022

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