A371715 G.f. A(x) = Sum_{n>=0} a(n)*x^n where a(n) = Sum_{k=0..n} ( [x^k] A(2*x)^(n+1) (mod 2^(n+1)) ) for n >= 0 with a(0) = 1.

1, 1, 7, 17, 55, 113, 135, 321, 2103, 3217, 8295, 18145, 33687, 74289, 128455, 321153, 870135, 1201105, 3696423, 6715937, 17466967, 31467889, 72111239, 173224385, 382697911, 552100625, 1442627047, 2558102881, 3487807767, 11651874993, 21616075591, 47612431617, 108598963319

Offset

0,3

Comments

What is the limit a(n)/(n*2^n) ? Does it exist ?. - Vaclav Kotesovec, May 03 2024

Links

Paul D. Hanna, Table of n, a(n) for n = 0..519

Vaclav Kotesovec, Plot of a(n)/(n*2^n)

Example

G.f.: A(x) = 1 + x + 7*x^2 + 17*x^3 + 55*x^4 + 113*x^5 + 135*x^6 + 321*x^7 + 2103*x^8 + 3217*x^9 + 8295*x^10 + 18145*x^11 + 33687*x^12 + ...
The table of coefficients of x^k in A(2*x)^n begins:
n=1: [1,  2,  28,  136,   880,   3616,    8640, ...];
n=2: [1,  4,  60,  384,  3088,  18368,   99520, ...];
n=3: [1,  6,  96,  752,  6960,  50592,  350848, ...];
n=4: [1,  8, 136, 1248, 12848, 107520,  864000, ...];
n=5: [1, 10, 180, 1880, 21120, 197312, 1765760, ...];
n=6: [1, 12, 228, 2656, 32160, 329088, 3210624, ...];
n=7: [1, 14, 280, 3584, 46368, 512960, 5383168, ...];
n=8: [1, 16, 336, 4672, 64160, 760064, 8500480, ...];
...
from which each term in row n may be reduced modulo 2^n to form the following triangle (trailing zeros suppressed):
A(2*x) (mod 2): [1];
A(2*x)^2 (mod 2^2): [1, 0];
A(2*x)^3 (mod 2^3): [1, 6, 0];
A(2*x)^4 (mod 2^4): [1, 8, 8, 0];
A(2*x)^5 (mod 2^5): [1, 10, 20, 24, 0];
A(2*x)^6 (mod 2^6): [1, 12, 36, 32, 32, 0];
A(2*x)^7 (mod 2^7): [1, 14, 24, 0, 32, 64, 0];
A(2*x)^8 (mod 2^8): [1, 16, 80, 64, 160, 0, 0, 0];
A(2*x)^9 (mod 2^9): [1, 18, 396, 296, 464, 224, 320, 384, 0];
A(2*x)^10 (mod 2^10): [1, 20, 460, 192, 624, 832, 64, 512, 512, 0];
A(2*x)^11 (mod 2^11): [1, 22, 528, 784, 80, 1120, 1664, 1536, 1536, 1024, 0];
A(2*x)^12 (mod 2^12): [1, 24, 600, 2592, 3920, 3328, 2048, 2048, 3584, 0, 0, 0];  ...
The row sums of the above triangle form this sequence.
SPECIFIC VALUES.
A(1/3) = 5.16342352287698185339110618005914219821...
A(1/4) = 2.39174482766273066991171836527389776948...
A(1/5) = 1.76139991234861303660669400163149230805...
A(1/6) = 1.50260759835187695870739455561470784874...
A(t) = 2 at t = 0.22255799946572991869110865358904969874...
A(t) = 3 at t = 0.27939565401912059198552904564135101228...
A(t) = 4 at t = 0.31035599522228672966119371322774606931...
A(t) = 5 at t = 0.33063501311279486918626289986794151717...

Prog

(PARI) /* Returns vector A of N terms */
{N = 40; A=vector(N); A[1]=1; for(n=2, #A, A[n]=1; A[n] = sum(k=0, n-1, ( polcoeff( Ser(A)^n, k)*2^k )%(2^n) ) ); A}

Crossrefs

Sequence in context: A352616 A094534 A262106 * A081632 A276907 A293464

Adjacent sequences: A371712 A371713 A371714 * A371716 A371717 A371718

Keyword

nonn

Author

Paul D. Hanna, Apr 30 2024

Status

approved