|
%I #38 Dec 09 2022 14:35:12
%S 1,3,7,33,163,858,4708,26662,154699,914885,5494719,33423598,205493244,
%T 1274928510,7972042450,50188844583,317861388939,2023777490895,
%U 12945901676736,83163975425669,536279878717858,3470134399230086,22525040920670333,146633283078321531
%N a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(2*n+1))^(n-1).
%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.
%H Paul D. Hanna, <a href="/A358961/b358961.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^(2*n+1))^(n-1).
%F (2) x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n^2) / (1 - x^(2*n-1)*A(x))^(n+1).
%F (3) A(x) = Sum_{n=-oo..+oo} x^(3*n+1)* (A(x) - x^(2*n+1))^(n-1).
%F (4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n*(n-1)) / (1 - x^(2*n-1)*A(x))^(n+1).
%F (5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n*(n-1)) / (1 - x^(2*n-1)*A(x))^n.
%e G.f.: A(x) = 1 + 3*x + 7*x^2 + 33*x^3 + 163*x^4 + 858*x^5 + 4708*x^6 + 26662*x^7 + 154699*x^8 + 914885*x^9 + 5494719*x^10 + ...
%e where A = A(x) satisfies the doubly infinite sum
%e 1 = ... + x^(-2)*(A - x^(-3))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x)^(-1) + x*(A - x^3)^0 + x^2*(A - x^5) + x^3*(A - x^7)^2 + x^4*(A - x^9)^3 + ... + x^n * (A - x^(2*n+1))^(n-1) + ...
%e also,
%e A(x) = ... + x^24/(1 - x^(-7)*A)^(-2) - x^12/(1 - x^(-5)*A)^(-1) + x^4 - 1/(1 - x^(-1)*A) + 1/(1 - x*A)^2 - x^4/(1 - x^3*A)^3 + x^12/(1 - x^5*A)^4 - x^24/(1 - x^7*A)^5 + ... + (-1)^(n+1)*x^(2*n*(n-1))/(1 - x^(2*n-1)*A(x))^(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^(2*n+1))^(n-1) ), #A-1) );A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A357227, A358962, A358963, A358964, A358965, A358937.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 07 2022
| 