A355868 G.f. A(x) satisfies: 1 = Sum_{n=-oo..+oo} (x^n - 2*x*A(x))^n.

1, 2, 3, 3, 5, 39, 206, 697, 1656, 3208, 8727, 41667, 192142, 688944, 1965643, 5117374, 15888133, 63924038, 263759291, 955198539, 3017571957, 9101208987, 30075674452, 113177783141, 437460265979, 1583161667787, 5299622270275, 17294182815347, 59169678008804

Offset: 0

Author

Paul D. Hanna, Aug 09 2022

Keywords

• nonn

Comments

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

Related identity: Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*x^n)^n = 0 for all y.

Examples

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 3*x^3 + 5*x^4 + 39*x^5 + 206*x^6 + 697*x^7 + 1656*x^8 + 3208*x^9 + 8727*x^10 + 41667*x^11 + 192142*x^12 + ...
where
1 = ... + (x^(-3) - 2*x*A(x))^(-3) + (x^(-2) - 2*x*A(x))^(-2) + (x^(-1) - 2*x*A(x))^(-1) + 1 + (x - 2*x*A(x)) + (x^2 - 2*x*A(x))^2 + (x^3 - 2*x*A(x))^3 + ... + (x^n - 2*x*A(x))^n + ...
and
1 = ... + x^(-5)/(x^(-3) + 2*A(x))^3 + x^(-3)/(x^(-2) + 2*A(x))^2 + x^(-1)/(x^(-1) + 2*A(x)) + x + x^3*(x + 2*A(x)) + x^5*(x^2 + 2*A(x))^2 + x^7*(x^3 + 2*A(x))^3 + ... + x^(2*n+1)*(x^n + 2*A(x))^n + ...
also,
1 = ... + x^9*(1 - 2*A(x)/x^2)^3 + x^4*(1 - 2*A(x)/x)^2 + x*(1 - 2*A(x)) + 1 + x/(1 - 2*A(x)*x^2) + x^4/(1 - 2*A(x)*x^3)^2 + x^9/(1 - 2*A(x)*x^4)^3 + ... + x^(n^2)/(1 - 2*A(x)*x^(n+1))^n + ...
further,
1 = ... + x^9*(1 + 2*A(x)/x^2)^2 + x^4*(1 + 2*A(x)/x) + x + 1/(1 + 2*A(x)*x) + x/(1 + 2*A(x)*x^2)^2 + x^4/(1 - 2*A(x)*x^3)^3 + x^9/(1 - 2*A(x)*x^4)^4 + ... + x^(n^2)/(1 + 2*A(x)*x^(n+1))^(n+1) + ...
SPECIFIC VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04720243920412572796492634515550526365563452970121157309...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 2*A)^n = 1.
(V.2) Let A = A(exp(-2*Pi)) = 0.001874436990256710694689538031391789940066981740061145959...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 2*A)^n = 1.
(V.3) Let A = A(-exp(-Pi)) = -0.03971121915244100584186154683625533541823516978831008865...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 2*A)^n = 1.
(V.4) Let A = A(-exp(-2*Pi)) = -0.001860487547859226152163099117755736250804492732905479139...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 2*A)^n = 1.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A355867, A359671, A359673.

Cf. A370041, A370030, A370031, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

Programs

PARI

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

PARI

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

PARI

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

PARI

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

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n=-oo..+oo} (x^n - 2*x*A(x))^n.

(2) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (x^n + 2*A(x))^n.

(3) 0 = Sum_{n=-oo..+oo} (-1)^n * (x^n - 2*x*A(x))^(n-1).

(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (x^n + 2*x*A(x))^(n+1).

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

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

(7) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^n)^n.

a(n) ~ c * d^n / n^(3/2), where d = 3.70839... and c = 1.176... - Vaclav Kotesovec, Feb 18 2024