VOOZH about

URL: https://oeis.org/A376179

⇱ A376179 - OEIS

login
A376179
a(n) = (1/2^n) * Sum_{k=0..2*n-1} ( binomial(2*n-1, k) (mod 2^n) ).
2
1, 2, 2, 2, 4, 4, 4, 6, 6, 8, 10, 8, 8, 8, 14, 12, 12, 14, 16, 12, 16, 12, 16, 18, 24, 22, 18, 24, 20, 22, 26, 22, 24, 28, 26, 28, 28, 28, 32, 32, 28, 26, 34, 26, 34, 40, 42, 38, 36, 32, 28, 36, 42, 48, 38, 46, 46, 40, 46, 42, 46, 50, 42, 50, 50, 48, 60, 56, 56, 54, 56, 48, 52, 58, 56, 58, 60, 64, 62, 60
OFFSET
1,2
COMMENTS
What is the limit of the average value of a(n)/n as n increases?
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Paul D. Hanna)
EXAMPLE
Term a(n) equals the sum of the residues of the binomial coefficients in (1 + x)^(2*n-1) modulo 2^n, divided by 2^n, as illustrated below.
a(1) = (1 + 1)/2 = 1;
a(2) = (1 + 3 + 3 + 1)/2^2 = 2;
a(3) = (1 + 5 + 2 + 2 + 5 + 1)/2^3 = 2;
a(4) = (1 + 7 + 5 + 3 + 3 + 5 + 7 + 1)/2^4 = 2;
a(5) = (1 + 9 + 4 + 20 + 30 + 30 + 20 + 4 + 9 + 1)/2^5 = 4;
a(6) = (1 + 11 + 55 + 37 + 10 + 14 + 14 + 10 + 37 + 55 + 11 + 1)/2^6 = 4;
a(7) = (1 + 13 + 78 + 30 + 75 + 7 + 52 + 52 + 7 + 75 + 30 + 78 + 13 + 1)/2^7 = 4;
a(8) = (1 + 15 + 105 + 199 + 85 + 187 + 141 + 35 + 35 + 141 + 187 + 85 + 199 + 105 + 15 + 1)/2^8 = 6;
a(9) = (1 + 17 + 136 + 168 + 332 + 44 + 88 + 504 + 246 + 246 + 504 + 88 + 44 + 332 + 168 + 136 + 17 + 1)/2^9 = 6;
a(10) = (1 + 19 + 171 + 969 + 804 + 364 + 508 + 212 + 830 + 218 + 218 + 830 + 212 + 508 + 364 + 804 + 969 + 171 + 19 + 1)/2^10 = 8;
...
SPECIFIC VALUES.
Sum_{n>=1} a(n)*(4/5)^n = 14.56731184692069354470343263208792452161116972...
Sum_{n>=1} a(n)*(3/4)^n = 8.806158075337438705490872541539624781638250680...
Sum_{n>=1} a(n)*(2/3)^n = 4.522622442349358863177945761420993373436343740...
Sum_{n>=1} a(n)/2^n = 1.645831621205319316132701685075584217399604974560493999848...
Sum_{n>=1} a(n)/3^n = 0.6795322549752092390181243171751347191553195443198...
Sum_{n>=1} a(n)/4^n = 0.4193145502805243188207226980231199521884988424193...
Sum_{n>=1} a(n)/5^n = 0.3008066971901188761765095346307808331285104623693...
PROG
(PARI) {a(n) = sum(k=0, 2*n-1, binomial(2*n-1, k) % (2^n) )/2^n}
for(n=1, 80, print1(a(n), ", "))
CROSSREFS
Sequence in context: A113694 A086159 A029048 * A086160 A029047 A007294
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 09 2024
STATUS
approved