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.

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

Paul D. Hanna, Table of n, a(n) for n = 0..300

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

Adjacent sequences: A357795 A357796 A357797 * A357799 A357800 A357801

Keyword

nonn

Author

Paul D. Hanna, Dec 22 2022

Status

approved