A358964 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(5*n+4))^(n-1).

1, 2, 7, 30, 144, 728, 3879, 21338, 120301, 691482, 4037020, 23873308, 142702222, 860823760, 5233702949, 32038319854, 197302553658, 1221511228130, 7598234842024, 47464203317986, 297630203452010, 1872792573164662, 11821420702394153, 74834134991237178

Offset: 0

Author

Paul D. Hanna, Dec 07 2022

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(5*n+4))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 144*x^4 + 728*x^5 + 3879*x^6 + 21338*x^7 + 120301*x^8 + 691482*x^9 + 4037020*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-6))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^4)^(-1) + x*(A - x^9)^0 + x^2*(A - x^14) + x^3*(A - x^19)^2 + x^4*(A - x^24)^3 + ... + x^n * (A - x^(5*n+4))^(n-1) + ...
also,
A(x) = ... + x^60/(1 - x^(-19)*A)^(-2) - x^30/(1 - x^(-14)*A)^(-1) + x^10 - 1/(1 - x^(-4)*A) + 1/(1 - x*A)^2 - x^10/(1 - x^6*A)^3 + x^30/(1 - x^11*A)^4 - x^60/(1 - x^16*A)^5 + ... + (-1)^(n+1)*x^(5*n*(n-1))/(1 - x^(5*n-4)*A)^(n+1) + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A358961, A358962, A358963, A358965.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n * (Ser(A) - x^(5*n+4))^(n-1) ), #A-1) );A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(5*n+4))^(n-1).

(2) x^4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(5*n^2) / (1 - x^(5*n-4)*A(x))^(n+1).

(3) A(x) = Sum_{n=-oo..+oo} x^(6*n+4)* (A(x) - x^(5*n+4))^(n-1).

(4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(5*n*(n-1)) / (1 - x^(5*n-4)*A(x))^(n+1).

(5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(5*n*(n-1)) / (1 - x^(5*n-4)*A(x))^n.