cp's OEIS Frontend

A358963 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(4*n+3))^(n-1).

%I A358963 #20 Dec 09 2022 14:33:41
%S A358963 1,2,7,31,143,731,3896,21444,120967,695699,4063879,24045306,143808836,
%T A358963 867972228,5280039896,32339575813,199266229047,1234340158837,
%U A358963 7682216027973,48014943810066,301247658649431,1896587278353158,11978138505184044,75867527248248561
%N A358963 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(4*n+3))^(n-1).
%C A358963 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(4*n+3))^n, which holds formally for all y.
%H A358963 Paul D. Hanna, <a href="/A358963/b358963.txt">Table of n, a(n) for n = 0..200</a>
%F A358963 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A358963 (1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(4*n+3))^(n-1).
%F A358963 (2) x^3 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(4*n^2) / (1 - x^(4*n-3)*A(x))^(n+1).
%F A358963 (3) A(x) = Sum_{n=-oo..+oo} x^(5*n+3)* (A(x) - x^(4*n+3))^(n-1).
%F A358963 (4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(4*n*(n-1)) / (1 - x^(4*n-3)*A(x))^(n+1).
%F A358963 (5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(4*n*(n-1)) / (1 - x^(4*n-3)*A(x))^n.
%e A358963 G.f.: A(x) = 1 + 2*x + 7*x^2 + 31*x^3 + 143*x^4 + 731*x^5 + 3896*x^6 + 21444*x^7 + 120967*x^8 + 695699*x^9 + 4063879*x^10 + ...
%e A358963 where A = A(x) satisfies the doubly infinite sum
%e A358963 1 = ... + x^(-2)*(A - x^(-5))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^3)^(-1) + x*(A - x^7)^0 + x^2*(A - x^11) + x^3*(A - x^15)^2 + x^4*(A - x^19)^3 + ... + x^n * (A - x^(4*n+3))^(n-1) + ...
%e A358963 also,
%e A358963 A(x) = ... + x^48/(1 - x^(-15)*A)^(-2) - x^24/(1 - x^(-11)*A)^(-1) + x^8 - 1/(1 - x^(-3)*A) + 1/(1 - x*A)^2 - x^8/(1 - x^5*A)^3 + x^24/(1 - x^9*A)^4 - x^48/(1 - x^13*A)^5 + ... + (-1)^(n+1)*x^(4*n*(n-1))/(1 - x^(4*n-3)*A)^(n+1) + ...
%o A358963 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A358963 A[#A] = polcoeff( sum(n=-#A,#A, x^n * (Ser(A) - x^(4*n+3))^(n-1) ), #A-1) );A[n+1]}
%o A358963 for(n=0,30,print1(a(n),", "))
%Y A358963 Cf. A358961. A358962, A358964, A358965.
%K A358963 nonn
%O A358963 0,2
%A A358963 _Paul D. Hanna_, Dec 07 2022