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A352276 a(0) = 1 and a(n) = Sum_{k = 0..3*n} n/(n + 2*k)*binomial(n + 2*k,k) for n >= 1. 2
1, 9, 625, 58785, 5986993, 633580634, 68611922731, 7545931449401, 839183314181297, 94112350842056469, 10623982584664109750, 1205644823097085684641, 137414820511792364274091, 15718880489100999321142976, 1803621273322664188151352631, 207499462144488863314062180035 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The following identity can be easily verified using Maple's SumTools:-Summation procedure: for n >= 1, A005810(n) = binomial(4*n,n) = Sum_{k = 0..3*n} n/(n + k)*binomial(n + k,k).

The binomial coefficients A005810(n) are known to satisfy the supercongruences A005810(n*p^r) == A005810(n*p^(r-1)) (mod p^(3*r)) for primes p >= 5 and positive integers n and r (see Meštrović, equation 39).

Calculation suggests that the present sequence satisfies the same congruences.

Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for primes p >= 5 and positive integers n and r.

More generally, for m a positive integer, define a sequence u_m by setting u_m(n) = Sum_{k = 0..m*n} n/(n + 2*k)*binomial(n + 2*k,k) for n >= 1.

Then we conjecture that each sequence u_m satisfies the above supercongruences. This is the case m = 3. See A333093 (case m = 1) and A352275 (m = 2).

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..475

R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2001.

FORMULA

a(n) ~ 7^(7*n + 3/2) / (37 * sqrt(Pi*n) * 2^(8*n + 3/2) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Mar 15 2022

EXAMPLE

Examples of supercongruences:

a(11) - a(1) = 1205644823097085684641 - 9 = (2^3)*3*(11^3)*43*2887*5059* 60096637 == 0 (mod 11^3)

a(3*5) - a(3) = 207499462144488863314062180035 - 58785 = 2*(5^4)*1801* 4959701*18583938263214197 == 0 (mod 5^4)

MATHEMATICA

nterms=25; Join[{1}, Table[Sum[n/(n+2k)Binomial[n+2k, k], {k, 0, 3n}], {n, nterms-1}]] (* Paolo Xausa, Apr 10 2022 *)

CROSSREFS

Cf. A005810, A333093, A352275.

Sequence in context: A226552 A277232 A171703 * A085530 A188423 A158881

Adjacent sequences:  A352273 A352274 A352275 * A352277 A352278 A352279

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Mar 10 2022

STATUS

approved

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Last modified September 25 18:27 EDT 2022. Contains 356986 sequences. (Running on oeis4.)