A351511 G.f. A(x) = Product_{n>=1} P(n,x), where P(1,x) = 1/sqrt(1-4*x), and P(n+1,x) = 1/sqrt(1 - 4*x + 4*x/P(n,x)) for n >= 1.

1, 2, 10, 40, 174, 696, 2932, 11824, 48630, 197080, 802028, 3244752, 13147372, 53097392, 214467688, 864956896, 3486971302, 14044699288, 56544883260, 227515358736, 915081831108, 3678931443984, 14785604430808, 59403722115616, 238598970849468, 958098090482416

Offset

0,2

Comments

The g.f. of this sequence is motivated by the (known) infinite product identity:

2/Pi = sqrt(1/2) * sqrt(1/2 + (1/2)*sqrt(1/2)) * sqrt(1/2 + (1/2)*sqrt(1/2 + (1/2)*sqrt(1/2))) * sqrt(1/2 + (1/2)*sqrt(1/2 + (1/2)*sqrt(1/2 + (1/2)*sqrt(1/2)))) * ...

Links

Table of n, a(n) for n=0..25.

Aaron Levin, A Geometric Interpretation of an Infinite Product for the Lemniscate Constant, The American Mathematical Monthly, vol. 113, no. 6, Mathematical Association of America, 2006, pp. 510-20.

Formula

A(1/8) = Pi/2.

Conjecture: a(n) ~ 4^n. - Vaclav Kotesovec, Feb 19 2022

Example

G.f.: A(x) = 1 + 2*x + 10*x^2 + 40*x^3 + 174*x^4 + 696*x^5 + 2932*x^6 + 11824*x^7 + 48630*x^8 + 197080*x^9 + 802028*x^10 + 3244752*x^11 + 13147372*x^12 + ...
where
A(x) = 1/sqrt(1-4*x) * 1/sqrt(1-4*x + 4*x*sqrt(1-4*x)) * 1/sqrt(1-4*x + 4*x*sqrt(1-4*x + 4*x*sqrt(1-4*x))) * 1/sqrt(1-4*x + 4*x*sqrt(1-4*x + 4*x*sqrt(1-4*x + 4*x*sqrt(1-4*x)))) * ...
Specific values:
A(1/8) = 1.5707963267948966192313... = Pi/2.
Related series:
log(A(x)) = 2*x + 16*x^2/2 + 68*x^3/3 + 320*x^4/4 + 1172*x^5/5 + 5152*x^6/6 + 19112*x^7/7 + 78848*x^8/8 + 306356*x^9/9 + 1227936*x^10/10 + ...
Related table:
The g.f. A(x) equals Product_{n>=1} P(n,x), where P(1,x) = 1/sqrt(1-4*x), and P(n+1,x) = 1/sqrt(1 - 4*x + 4*x/P(n,x)) for n >= 1.
The table of coefficients of x^k in P(n,x) begins:
n=1: [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620,  ...];
n=2: [1, 0, 4,  4, 32,  68, 336,  984,  4096, 13844, 54160, ...];
n=3: [1, 0, 0,  8,  8,  32, 168,  448,  1616,  6400, 21864, ...];
n=4: [1, 0, 0,  0, 16,  16,  64,  208,  1024,  2848, 10880, ...];
n=5: [1, 0, 0,  0,  0,  32,  32,  128,   416,  1536,  6208, ...];
n=6: [1, 0, 0,  0,  0,   0,  64,   64,   256,   832,  3072, ...];
n=7: [1, 0, 0,  0,  0,   0,   0,  128,   128,   512,  1664, ...];
n=8: [1, 0, 0,  0,  0,   0,   0,    0,   256,   256,  1024, ...];
...
Limit_{n->infinity} (P(n) - 1)/(2*x)^n = 1 + x + 4*x^2 + 13*x^3 + 48*x^4 + 162*x^5 + 600*x^6 + 2109*x^7 + 7760*x^8 + 28166*x^9 + ... + A351509(n)*x^n + ...

Mathematica

P[1, n2_] = 1/Sqrt[1 - 4*x + x*O[x]^n2]; P[n1_, n2_] := 1/Sqrt[1 - 4*x + 4*x/P[n1 - 1, n2] + x*O[x]^n2]; CoefficientList[Product[P[n1, # - 1], {n1, 1, #}], x] &[26] (* Robert P. P. McKone, Feb 13 2022 *)

Prog

(PARI) N = 30 ; \\ number of desired terms
{a(n) = my(P = vector(N+1)); P[1] = 1/sqrt(1 - 4*x +x*O(x^N)); for(n=1, N,
P[n+1] = 1/sqrt(1 - 4*x + 4*x/P[n] +x*O(x^N) )); Vec(prod(n=1, N+1, P[n]))[n+1]}
for(n=0, N, print1(a(n), ", "))

Crossrefs

Cf. A351509, A351512.

Sequence in context: A390683 A223095 A052978 * A151023 A344501 A151024

Adjacent sequences: A351508 A351509 A351510 * A351512 A351513 A351514

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2022

Status

approved