A351512 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, 2, 8, 30, 120, 452, 1648, 5910, 21592, 81084, 309264, 1178956, 4476208, 17010568, 64978016, 249438182, 960341912, 3703401484, 14301548880, 55314019492, 214292575376, 831516584504, 3230994776480, 12569343719036, 48949513156080, 190818329097816

Offset

0,2

Comments

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

2/L = 2/A062539 = 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)))) * ..., where L = A062539 is the Lemniscate constant.

Links

Table of n, a(n) for n=0..26.

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) = L/2, where L = gamma(1/4)^2/sqrt(8*Pi) = 2.622057554292... is the Lemniscate constant.

Example

G.f.: A(x) = 1 + 2*x + 2*x^2 + 8*x^3 + 30*x^4 + 120*x^5 + 452*x^6 + 1648*x^7 + 5910*x^8 + 21592*x^9 + 81084*x^10 + 309264*x^11 + 1178956*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) = 1.3110287771460599052324... = L/2, where L is the Lemniscate constant (cf. A062539).
Related series:
log(A(x)) = 2*x + 20*x^3/3 + 64*x^4/4 + 372*x^5/5 + 1440*x^6/6 + 5896*x^7/7 + 22016*x^8/8 + 87284*x^9/9 + 352160*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, -12, -16,   4,  32, -168, -1376, -5100, ...];
n=3: [1, 0,  0,   8,  24,  32,  88,  512,  1968,  6144, ...];
n=4: [1, 0,  0,   0, -16, -48, -64, -176,  -640, -1632, ...];
n=5: [1, 0,  0,   0,   0,  32,  96,  128,   352,  1280, ...];
n=6: [1, 0,  0,   0,   0,   0, -64, -192,  -256,  -704, ...];
n=7: [1, 0,  0,   0,   0,   0,   0,  128,   384,   512, ...];
n=8: [1, 0,  0,   0,   0,   0,   0,    0,  -256,  -768, ...];
...
Limit_{n->infinity} (1 - P(n))/(-2*x)^n = 1 + 3*x + 4*x^2 + 11*x^3 + 40*x^4 + 150*x^5 + 552*x^6 + 1971*x^7 + 6952*x^8 + 24818*x^9 + ... + A351510(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] &[27] (* 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. A351510, A062539, A351511.

Sequence in context: A003616 A276657 A079895 * A053047 A076143 A226585

Adjacent sequences: A351509 A351510 A351511 * A351513 A351514 A351515

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2022

Status

approved