A366229 -id:A366229 - cp's OEIS frontend

A355866 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).

1, 2, 5, 20, 77, 319, 1357, 5861, 25934, 117970, 554949, 2713732, 13801721, 72690859, 393319668, 2166067444, 12036890380, 67038139970, 372431798808, 2058011292264, 11296150608376, 61573508814470, 333509165576785, 1797289086416868, 9653137938138051

Offset: 0

Paul D. Hanna, Aug 04 2022

nonn

Compare to the identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n which holds for all y.

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 77*x^4 + 319*x^5 + 1357*x^6 + 5861*x^7 + 25934*x^8 + 117970*x^9 + 554949*x^10 + 2713732*x^11 + ...
where
0 = ... + x^(-3)/(x^(-3) - A(x))^8 + x^(-2)/(x^(-2) - A(x))^5 + x^(-1)/(x^(-1) - A(x))^2 + (1 - A(x)) + x*(x - A(x))^4 + x^2*(x^2 - A(x))^7 + x^3*(x^3 - A(x))^10 + ... + x^n * (x^n - A(x))^(3*n+1) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A355865, A366229.

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n*(x^n - Ser(A))^(3*n+1) ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(n*(3*n-2)) / (1 - A(x)*x^n)^(3*n-1).

A358937 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^n.

Original entry on oeis.org

1, 1, 3, 6, 13, 31, 76, 192, 504, 1351, 3668, 10082, 27991, 78335, 220778, 626141, 1785593, 5117179, 14729826, 42568767, 123465517, 359268141, 1048541699, 3068583485, 9002849260, 26474484680, 78019959584, 230381635121, 681544367457, 2019718168994, 5995000501189

Offset: 0

Paul D. Hanna, Dec 07 2022

nonn

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 31*x^5 + 76*x^6 + 192*x^7 + 504*x^8 + 1351*x^9 + 3668*x^10 + 10082*x^11 + 27991*x^12 + ...

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

Cf. A366229, A355865.

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n * (x^(2*n) - Ser(A))^n ), #A) );A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(n*(2*n-1)) / (1 - x^(2*n)*A(x))^n.
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^(2*n+1) - A(x))^n (trivial).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 - x^(2*n)*A(x))^n (trivial).

cp's OEIS Frontend

A355866 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).

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