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URL: https://oeis.org/A357798

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A357798
a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^(n+1) * (2 - x^(n+1))^n * A(x)^n.
2
1, 2, 6, 20, 78, 364, 1758, 9144, 48508, 264014, 1457624, 8158260, 46134878, 263312552, 1514534512, 8771202984, 51101608190, 299306977508, 1761377916048, 10409550718692, 61755225688926, 367639850029404, 2195551697108888, 13149811270786752, 78967249613057836, 475373797733460598
OFFSET
0,2
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, 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} x^(n+1) * (2 - x^(n+1))^n * A(x)^n.
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / ((1 - 2*x^n)^(n+1) * A(x)^(n+1)).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 78*x^4 + 364*x^5 + 1758*x^6 + 9144*x^7 + 48508*x^8 + 264014*x^9 + 1457624*x^10 + 8158260*x^11 + 46134878*x^12 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^(n+1) * (2 - x^(n+1) +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); 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, #A, (-1)^(n+1) * x^(n^2) / ((1 - 2*x^n +x*O(x^#A) )^(n+1) * Ser(A)^(n+1)) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A357797.
Sequence in context: A098469 A081563 A038393 * A027221 A346747 A300514
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 22 2022
STATUS
approved