A358964 - OEIS

%I #18 Dec 09 2022 14:34:19

%S 1,2,7,30,144,728,3879,21338,120301,691482,4037020,23873308,142702222,

%T 860823760,5233702949,32038319854,197302553658,1221511228130,

%U 7598234842024,47464203317986,297630203452010,1872792573164662,11821420702394153,74834134991237178

%N 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).

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

%H Paul D. Hanna, <a href="/A358964/b358964.txt">Table of n, a(n) for n = 0..200</a>

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

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

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

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

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

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

%e 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 + ...

%e where A = A(x) satisfies the doubly infinite sum

%e 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) + ...

%e also,

%e 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) + ...

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

%o A[#A] = polcoeff( sum(n=-#A,#A, x^n * (Ser(A) - x^(5*n+4))^(n-1) ), #A-1) );A[n+1]}

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

%Y Cf. A358961, A358962, A358963, A358965.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 07 2022