A358965 - OEIS

%I #16 Dec 09 2022 14:34:47

%S 1,2,7,30,143,729,3876,21321,120195,690816,4032807,23846485,142530516,

%T 859719414,5226571568,31992109155,197002217763,1219554190530,

%U 7585453430037,47380560231549,297081856642195,1869191995298989,11797744585161792,74678247991840230,473954364916279312

%N a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(6*n+5))^(n-1).

%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(6*n+5))^n, which holds formally for all y.

%H Paul D. Hanna, <a href="/A358965/b358965.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^(6*n+5))^(n-1).

%F (2) x^5 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(6*n^2) / (1 - x^(6*n-5)*A(x))^(n+1).

%F (3) A(x) = Sum_{n=-oo..+oo} x^(7*n+5)* (A(x) - x^(6*n+5))^(n-1).

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

%F (5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(6*n*(n-1)) / (1 - x^(6*n-5)*A(x))^n.

%e G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 729*x^5 + 3876*x^6 + 21321*x^7 + 120195*x^8 + 690816*x^9 + 4032807*x^10 + ...

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

%e 1 = ... + x^(-2)*(A - x^(-7))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^5)^(-1) + x*(A - x^11)^0 + x^2*(A - x^17) + x^3*(A - x^23)^2 + x^4*(A - x^29)^3 + ... + x^n * (A - x^(6*n+5))^(n-1) + ...

%e also,

%e A(x) = ... + x^72/(1 - x^(-23)*A)^(-2) - x^36/(1 - x^(-17)*A)^(-1) + x^12 - 1/(1 - x^(-5)*A) + 1/(1 - x*A)^2 - x^12/(1 - x^7*A)^3 + x^36/(1 - x^13*A)^4 - x^72/(1 - x^19*A)^5 + ... + (-1)^(n+1)*x^(6*n*(n-1))/(1 - x^(6*n-5)*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^(6*n+5))^(n-1) ), #A-1) );A[n+1]}

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

%Y Cf. A358961, A358962, A358963, A358964.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 07 2022