A357791 -id:A357791 - cp's OEIS frontend

A359672 a(n) = coefficient of x^n in A(x) where x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n.

1, 1, 2, 5, 21, 72, 257, 998, 3988, 16064, 65734, 273541, 1151184, 4886946, 20916523, 90181047, 391230537, 1706503782, 7480000600, 32930469730, 145546039760, 645574246834, 2872745389578, 12821285282360, 57377599801569, 257416078950987, 1157519956026736, 5216112572700566

Offset: 0

Paul D. Hanna, Jan 10 2023

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 21*x^4 + 72*x^5 + 257*x^6 + 998*x^7 + 3988*x^8 + 16064*x^9 + 65734*x^10 + 273541*x^11 + 1151184*x^12 + ...
where
x = ... + x^6*A(x)^9/(1 + x^3*A(x)^3)^3 - x^2*A(x)^4/(1 + x^2*A(x)^2)^2 + A(x)/(1 + x*A(x)) - 1 + x*(1 + x*A(x)) - x^2*(1 + x^2*A(x)^2)^2 + x^3*(1 + x^3*A(x)^3)^3 + ... + (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n + ...
SPECIFIC VALUES.
A(1/d) = 1.71831164... where d = 4.76347639696677679... is given in the formula section.
A(1/5) = 1.47621312973364884841150188176844829427560286588046...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A357791, A357399, A357797.

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(n*(n-1)) * A(x)^(n^2) / (1 + x^n*A(x)^n)^n.
a(n) ~ c * d^n / n^(3/2), where d = 4.76347639696677679... and c = 0.37393658540119283... - Vaclav Kotesovec, Jan 11 2023

A357399 Coefficients of x^n, n >= 0, in A(x) such that: x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 3, 10, 37, 143, 564, 2270, 9305, 38755, 163569, 698186, 3009129, 13077850, 57250728, 252221229, 1117409653, 4975095073, 22249463540, 99901607730, 450187852401, 2035353779794, 9229671434155, 41968536407303, 191318458136066, 874179701912764, 4002949886221529

Offset: 0

Paul D. Hanna, Nov 07 2022

nonn

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

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 143*x^5 + 564*x^6 + 2270*x^7 + 9305*x^8 + 38755*x^9 + 163569*x^10 + 698186*x^11 + 3009129*x^12 + ...
such that A = A(x) satisfies
x = ... + (x*A)^(-2)/(1 - x^(-2))^2 - (x*A)^(-1)/(1 + x^(-1)) + 1 - (x*A)*(1 + x) + (x*A)^2*(1 - x^2)^2 - (x*A)^3*(1 + x^3)^3 + (x*A)^4*(1 - x^4)^4 + ... + (-x*A)^n*(1 - (-x)^n)^n + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.205304925829036018515173714762264122490105588782437881...
A(1/5) = 1.772537703863275044370749593488314266794544517965356320...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A357791.

{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(x - sum(n=-#A,#A, (-x)^n * (1 - (-x)^n +x*O(x^#A))^n * Ser(A)^n ),#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.
(1) x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 - (-x)^n)^n * A(x)^n).

cp's OEIS Frontend

A359672 a(n) = coefficient of x^n in A(x) where x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (1 + x^n*A(x)^n)^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula