OFFSET
0,2
COMMENTS
The g.f. of this sequence is motivated by the (known) infinite product identity:
LINKS
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 12 2022
STATUS
approved