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A173774 The arithmetic mean of (21k+8)*binomial(2k,k)^3 (k=0..n-1). 12
8, 120, 3680, 144760, 6427008, 306745824, 15364514880, 796663553400, 42395640372800, 2302336805317120, 127078484504270208, 7108177964254013920, 402042028998035350400, 22954860061516225396800 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

On Feb 10 2010, Zhi-Wei Sun introduced the sequence and conjectured that each term a(n) is an integer divisible by 4*binomial(2n,n). On Feb 11 2011, _Kasper Andersen_ confirmed this conjecture by noting that the sequence b(n) = a(n)/(4*binomial(2n,n)) (n=1,2,3,...) coincides with A112029. Zhi-Wei Sun proved that for every prime p and positive integer a we have a(p^a) == 8 + 1*6p^3*B_(p-3) (mod p^4), where B_0, B_1, B_2, ... are Bernoulli numbers. Given a prime p, Zhi-Wei Sun conjectured that Sum_{k=0..(p-1)/2} (21k+8)*binomial(2k,k)^3 == 8p + (-1)^((p-1)/2)*32*p^3*E_(p-3) (mod p^4) if p > 3 (where E_0, E_1, E_2, ... are Euler numbers), and that Sum_{k=0..floor(2p^a/3)} (21k+8)*binomial(2k,k)^3 == 8p^a (mod p^(a + 5 + (-1)^p)) if a is a positive integer with p^a == 1 (mod 3). He also observed that b(n) = a(n)/(4*binomial(2n,n)) is odd if and only if n is a power of two.

LINKS

Table of n, a(n) for n=1..14.

Kasper Andersen, Re: A somewhat surprising conjecture

Zhi-Wei Sun, A somewhat surprising conjecture

Zhi-Wei Sun, Re: A somewhat surprising conjecture

Zhi-Wei Sun, Open conjectures on congruences, preprint, arXiv:0911.5665 [math.NT], 2009-2011.

Zhi-Wei Sun, Super congruences and Euler numbers, preprint, arXiv:1001.4453 [math.NT], 2010-2011.

FORMULA

a(n) = (1/n)*Sum_{k=0..n-1} (21k+8)*binomial(2k,k)^3. Also, a(1)=8 and (n+1)*a(n+1) = n*a(n) + 8*(21n+8)*binomial(2n-1,n)^3 (n=1,2,3,...).

a(n) ~ 2^(6*n) / (3 * (Pi*n)^(3/2)). - Vaclav Kotesovec, Jan 24 2019

EXAMPLE

For n=2 we have a(2)=120 since (8*binomial(0,0)^3 + (21+8)*binomial(2,1)^3)/2 = 120.

MATHEMATICA

SS[n_]:=Sum[(21k+8)Binomial[2k, k]^3, {k, 0, n-1}]/n Table[SS[n], {n, 1, 25}]

CROSSREFS

Cf. A112029, A000984, A122045.

Sequence in context: A218671 A045754 A229045 * A034669 A000848 A214426

Adjacent sequences:  A173771 A173772 A173773 * A173775 A173776 A173777

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 24 2010

STATUS

approved

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Last modified August 10 07:37 EDT 2020. Contains 336368 sequences. (Running on oeis4.)