cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

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Author

Paul D. Hanna, Dec 03 2004

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

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

Programs

  • 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),", "))

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.

Extensions

Entry revised by Paul D. Hanna, Mar 05 2024

A101190 G.f.: A(x) = Sum_{n>=0} a(n)/2^A005187(n) * x^n = lim_{n->oo} F(n)^(1/2^n) where F(n) is defined by F(n) = F(n-1)^2 + x^(2^n-1) for n >= 1 with F(0) = 1.

Original entry on oeis.org

1, 1, -1, 5, -53, 127, -677, 2221, -61133, 205563, -1394207, 4852339, -68586849, 243751723, -1741612525, 6265913725, -363239625661, 1323861506899, -9699189175227, 35700526467479, -527987675255931, 1960112858076289, -14606721595781139, 54604708004873403
Offset: 0

Views

Author

Paul D. Hanna, Dec 03 2004

Keywords

Examples

			G.f.: A(x) = 1 + 1/2*x - 1/8*x^2 + 5/16*x^3 - 53/128*x^4 + 127/256*x^5 - 677/1024*x^6 + 2221/2048*x^7 + ... + a(n)/2^A005187(n)*x^n + ...
where 2^A005187(n) is also the denominator of [x^n] 1/sqrt(1-x).
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 + x^(2^1-1) = 1 + x,
F(2) = F(1)^2 + x^(2^2-1) = 1 + 2*x + x^2 + x^3,
F(3) = F(2)^2 + x^(2^3-1) = 1 + 4*x + 6*x^2 + 6*x^3 + 5*x^4 + 2*x^5 + x^6 + x^7.
...
The 2^n-th roots of F(n) tend to the limit of the g.f.:
F(1)^(1/2^1) = 1 + 1/2*x - 1/8*x^2 + 1/16*x^3 - 5/128*x^4 + 7/256*x^5 - 21/1024*x^6 + 33/2048*x^7 - 429/32768*x^8 + ...
F(2)^(1/2^2) = 1 + 1/2*x - 1/8*x^2 + 5/16*x^3 - 53/128*x^4 + 127/256*x^5 - 677/1024*x^6 + 1965/2048*x^7 - 46797/32768*x^8 + ...
F(3)^(1/2^3) = 1 + 1/2*x - 1/8*x^2 + 5/16*x^3 - 53/128*x^4 + 127/256*x^5 - 677/1024*x^6 + 2221/2048*x^7 - 61133/32768*x^8 + ...
...
The limit of this process equals the g.f. A(x) of this sequence.
Note: the sum of the coefficients in F(n) equals A003095(n):
1, 2 = 1 + 1, 5 = 1 + 2 + 1 + 1, 26 = 1 + 4 + 6 + 6 + 5 + 2 + 1 + 1, ...
The last n coefficients in F(n) read backwards are Catalan numbers (A000108).
POWERS OF A(x).
The coefficients of x^k in the 2^n powers of the g.f. A(x) begin:
A^(2^0) = [1, 1/2, -1/8, 5/16, -53/128, 127/256, -677/1024, 2221/2048, ...],
A^(2^1) = [1, 1, 0, 1/2, -1/2, 1/2, -5/8, 9/8, -2, 53/16, -89/16, 155/16, ...],
A^(2^2) = [1, 2, 1, 1, 0, 0, 0, 1/2, -1, 3/2, -5/2, 9/2, -8, 14, -197/8, 44, ...],
A^(2^3) = [1, 4, 6, 6, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1/2, -2, 5, ...],
A^(2^4) = [1, 8, 28, 60, 94, 116, 114, 94, 69, 44, 26, 14, 5, 2, 1, 1, 0, 0, ...].

Crossrefs

Cf. A101189, A101191, A005187, A003095, A000108.

Programs

  • 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 + x^(2^k-1) +x*O(x^n)); A = polcoeff(F^(1/2^L),n)); numerator(A)}
for(n=0,32, print1(a(n),", "))

Formula

G.f. A(x) = ( Sum_{n>=0} A101191(n)/2^A004134(n) * x^n )^2.
G.f. A(x) satisfies A(2*x)^2 = Sum_{n>=0} A101189(n)*(2*x)^n.

Extensions

Entry revised by Paul D. Hanna, Mar 05 2024
Showing 1-2 of 2 results.

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