A351509 G.f. A(x) = lim_{n->infinity} (P(n) - 1)/(2*x)^n, 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, 1, 4, 13, 48, 162, 600, 2109, 7760, 28166, 103992, 383218, 1427312, 5317284, 19934768, 74908957, 282588832, 1068648734, 4053382328, 15408585766, 58713314864, 224166303708, 857527923728, 3286072863410, 12613110808416, 48486435978236

Offset

0,3

Comments

Is there a simpler formula for the g.f. of this sequence?

The g.f. of A351511 equals Product_{n>=1} P(n,x), where P(n,x) is defined in the example section of this sequence, with a special value found at x = 1/8: Product_{n>=1} P(n,x=1/8) = Pi/2.

Odd terms occur at positions 2^n - 1 for n >= 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..520

Formula

A(1/8) = Pi^2/8 = 1.2337005501361698...

a(2^n - 1) = 1 (mod 2) for n >= 0.

Conjecture: a(n) ~ 4^n/n. - Vaclav Kotesovec, Mar 05 2022

Example

G.f.: A(x) = 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 + 103992*x^10 + 383218*x^11 + 1427312*x^12 + ...
Related tables.
The functions P(n,x) start with P(1,x) = 1/sqrt(1-4*x), and continue as 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, 184756, ...];
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, ...];
...
from which we can see that the coefficients in (P(n,x) - 1)/(2*x)^n tend to converge to this sequence as a limit:
n=1: [1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, ...];
n=2: [1, 1,  8, 17,  84, 246, 1024, 3461, 13540, 48842, ...];
n=3: [1, 1,  4, 21,  56, 202,  800, 2733, 10080, 37870, ...];
n=4: [1, 1,  4, 13,  64, 178,  680, 2381,  9136, 32422, ...];
n=5: [1, 1,  4, 13,  48, 194,  632, 2269,  8304, 30406, ...];
n=6: [1, 1,  4, 13,  48, 162,  664, 2173,  8080, 29254, ...];
n=7: [1, 1,  4, 13,  48, 162,  600, 2237,  7888, 28806, ...];
n=8: [1, 1,  4, 13,  48, 162,  600, 2109,  8016, 28422, ...];
n=9: [1, 1,  4, 13,  48, 162,  600, 2109,  7760, 28678, ...];
...

Prog

(PARI) {a(n) = my(P = 1/sqrt(1 - 4*x +x*O(x^(2*n+1)))); for(i=1, n,
P = 1/sqrt(1 - 4*x + 4*x/P +x*O(x^(2*n+1))) ); polcoeff( (P-1)/(2*x)^(n+1), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A351511, A351510.

Sequence in context: A149443 A125656 A279159 * A149444 A149445 A149446

Adjacent sequences: A351506 A351507 A351508 * A351510 A351511 A351512

Keyword

nonn

Author

Paul D. Hanna, Feb 14 2022

Status

approved