A326284 G.f. A(x) satisfies: 1 = Sum_{n>=0} 4^n * ((1+x)^n - A(x))^n.

1, 1, 4, 104, 4196, 225216, 14845072, 1151255440, 102289538128, 10226417550096, 1135388485042624, 138583671424928128, 18446474604149746176, 2659732597343823233280, 413060592233577210697984, 68754628660531280009195776, 12213125156726936259944672320, 2306358043375070604869802287616, 461443265563759624969778550969344, 97514484569091438266511351355560448

Offset

0,3

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,

(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),

for any fixed integer k; here, k = 1 with r = 4, p = -A(x), q = (1+x).

Links

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

Formula

G.f. A(x) satisfies:

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

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

a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.034391206985341... - Vaclav Kotesovec, Oct 13 2020

Example

G.f.: A(x) = 1 + x + 4*x^2 + 104*x^3 + 4196*x^4 + 225216*x^5 + 14845072*x^6 + 1151255440*x^7 + 102289538128*x^8 + 10226417550096*x^9 + 1135388485042624*x^10 + ...
such that
1 = 1  +  4*((1+x) - A(x))  +  4^2*((1+x)^2 - A(x))^2  +  4^3*((1+x)^3 - A(x))^3  +  4^4*((1+x)^4 - A(x))^4  +  4^5*((1+x)^5 - A(x))^5  +  4^6*((1+x)^6 - A(x))^6  +  4^7*((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + 4*A(x))  +  4*(1+x)/(1 + 4*(1+x)*A(x))^2  +  4^2*(1+x)^4/(1 + 4*(1+x)^2*A(x))^3  +  4^3*(1+x)^9/(1 + 4*(1+x)^3*A(x))^4  +  4^4*(1+x)^16/(1 + 4*(1+x)^4*A(x))^5  +  4^5*(1+x)^25/(1 + 4*(1+x)^5*A(x))^6  +  4^6*(1+x)^36/(1 + 4*(1+x)^6*A(x))^7 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, 4^m*((1+x)^m - Ser(A))^m ) )[#A]/4 ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A303056, A326282, A326283.

Sequence in context: A196979 A197164 A303135 * A302733 A098696 A013054

Adjacent sequences: A326281 A326282 A326283 * A326285 A326286 A326287

Keyword

nonn

Author

Paul D. Hanna, Jun 22 2019

Status

approved