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URL: https://oeis.org/A357795

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A357795
a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2).
3
1, 4, 26, 300, 4134, 61696, 969660, 15837400, 266125823, 4571229248, 79904206064, 1416736880104, 25418030469904, 460600399886240, 8417980252615072, 154985730303047328, 2871904782258356719, 53519211809275995362, 1002383232008661189884, 18858606600633628740774
OFFSET
0,2
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (y - x^n)^(n-1), which holds formally for all y.
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2).
(2) 1 = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)/3! * x^(n*(n-3)) / ((1 - x^(n-2))^n * A(x)^(n-2)).
EXAMPLE
G.f.: A(x) = 1 + 4*x + 26*x^2 + 300*x^3 + 4134*x^4 + 61696*x^5 + 969660*x^6 + 15837400*x^7 + 266125823*x^8 + 4571229248*x^9 + 79904206064*x^10 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-1, #A+1, n*(n+1)*(n+2)/3! * x^n * if(n==-2, 0, (1 - x^(n+2) +x*O(x^#A) )^n) * Ser(A)^(n+2) ), #A-1) ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-1, #A+1, (-1)^(n-1) * n*(n-1)*(n-2)/3! * x^(n*(n-3)) * if(n==2, 0, 1/(1 - x^(n-2) +x*O(x^#A) )^n) / Ser(A)^(n-2) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 22 2022
STATUS
approved