A101189 G.f. A(x) is defined as the limit A(x) = lim_{n->oo} F(n)^(1/2^(n-1)) where F(n) is defined by F(n) = F(n-1)^2 + (2*x)^(2^n-1) for n >= 1 with F(0) = 1.

1, 2, 0, 4, -8, 16, -40, 144, -512, 1696, -5696, 19840, -70048, 247744, -880128, 3152768, -11386624, 41389568, -151273728, 555794944, -2052141056, 7610274816, -28331018240, 105833345024, -396594444800, 1490425179136, -5615651143680, 21209004267520, -80276663808000

Offset

0,2

Comments

Sequences A101190 and A101191 are related to doubly exponential numbers A003095 and to Catalan numbers (A000108).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1025

Formula

G.f. A(x) = ( Sum_{n>=0} A101190(n)/2^A005187(n) * (2*x)^n )^2.

G.f. A(x) = ( Sum_{n>=0} A101191(n)/2^A004134(n) * (2*x)^n )^4.

Example

G.f.: A(x) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 144*x^7 - 512*x^8 + 1696*x^9 - 5696*x^10 + 19840*x^11 - 70048*x^12 + ...
GENERATING METHOD.
We can illustrate the generating method for g.f. A(x) as follows.
Given F(n) = F(n-1)^2 + (2*x)^(2^n-1) for n >= 1 with F(0) = 1,
the first few polynomials generated by F(n) begin
F(0) = 1,
F(1) = F(0)^2 + (2*x)^(2^1-1) = 1 + 2*x,
F(2) = F(1)^2 + (2*x)^(2^2-1) = 1 + 4*x + 4*x^2 + 8*x^3,
F(3) = F(2)^2 + (2*x)^(2^3-1) = 1 + 8*x + 24*x^2 + 48*x^3 + 80*x^4 + 64*x^5 + 64*x^6 + 128*x^7,
F(4) = F(3)^2 + (2*x)^(2^4-1) = = 1 + 16*x + 112*x^2 + 480*x^3 + 1504*x^4 + 3712*x^5 + 7296*x^6 + 12032*x^7 + 17664*x^8 + 22528*x^9 + 26624*x^10 + 28672*x^11 + 20480*x^12 + 16384*x^13 + 16384*x^14 + 32768*x^15,
...
and the 2^(n-1)-th root of F(n) yields the series shown by
F(1)^(1/2^0) = 1 + 2*x,
F(2)^(1/2^1) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 112*x^7 - 320*x^8 + 928*x^9 - 2752*x^10 + 8320*x^11 - 25504*x^12 + ...,
F(3)^(1/2^2) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 144*x^7 - 512*x^8 + 1696*x^9 - 5696*x^10 + 19840*x^11 - 70048*x^12 + ...,
F(4)^(1/2^3) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 144*x^7 - 512*x^8 + 1696*x^9 - 5696*x^10 + 19840*x^11 - 70048*x^12 + ...,
...
The limit of this process tends to the g.f. A(x).

Prog

(PARI) {a(n) = my(F=1, A, L); if(n==0, A=1, L = ceil(log(n+1)/log(2)); for(k=1, L, F = F^2 + (2*x)^(2^k-1)  +x*O(x^n)); A = polcoeff(F^(1/(2^(L-1))), n)); A}
for(n=0, 32, print1(a(n), ", "))

Crossrefs

Cf. A101190, A101191, A005187, A004134, A003095.

Sequence in context: A120554 A120710 A115780 * A295321 A372002 A366363

Adjacent sequences: A101186 A101187 A101188 * A101190 A101191 A101192

Keyword

sign

Author

Paul D. Hanna, Dec 03 2004

Extensions

Entry revised by Paul D. Hanna, Mar 05 2024

Status

approved