The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A333096

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A333096

a(n) = the n-th order Taylor polynomial (centered at 0) of c(x)^(4*n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the sequence of Catalan numbers A000108.
(history; published version)

#17 by N. J. A. Sloane at Fri May 03 15:07:20 EDT 2024

STATUS

proposed

approved

#16 by Peter Bala at Fri May 03 10:16:36 EDT 2024

STATUS

editing

proposed

#15 by Peter Bala at Fri May 03 09:38:23 EDT 2024

COMMENTS

We conjecture that the sequence satisfies the stronger congruences supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. Examples of these congruences are given below.

More generally, for each integer m, we conjecture that the sequence a_m(n) := the n-th order Taylor polynomial of c(x)^(m*n) evaluated at x = 1 satisfies the same congruencessupercongruences. For cases see A099837 (m = -2), A100219 (m = -1), A000012 (m = 0), A333093 (m = 1), A333094 (m = 2), A333095 (m = 3), A333097 (m = 5).

FORMULA

a(n) = Sum_{k = 0..n} 4*n/(4*n+k)*binomial(4*n+2*k-1, k) for n >= 1.

a(n) = Sum_{k = 0..n} 4*n/(4*n+2*k)*binomial(4*n+2*k, k) for n >= 1. - Peter Bala, May 03 2024

STATUS

approved

editing

#14 by N. J. A. Sloane at Wed Oct 06 14:25:16 EDT 2021

STATUS

editing

approved

#13 by N. J. A. Sloane at Wed Oct 06 14:25:14 EDT 2021

COMMENTS

We conjecture that the sequence satisfies the stronger supercongruences congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. Examples of these congruences are given below.

More generally, for each integer m, we conjecture that the sequence a_m(n) := the n-th order Taylor polynomial of c(x)^(m*n) evaluated at x = 1 satisfies the same supercongruencescongruences. For cases see A099837 (m = -2), A100219 (m = -1), A000012 (m = 0), A333093 (m = 1), A333094 (m = 2), A333095 (m = 3), A333097 (m = 5).

STATUS

approved

editing

#12 by N. J. A. Sloane at Wed Oct 06 14:24:32 EDT 2021

EXAMPLE

Examples of supercongruencescongruences:

Discussion

Wed Oct 06

14:24

OEIS Server: https://oeis.org/edit/global/2912

#11 by Vaclav Kotesovec at Sat Mar 28 07:35:40 EDT 2020

STATUS

editing

approved

#10 by Vaclav Kotesovec at Sat Mar 28 07:35:30 EDT 2020

FORMULA

a(n) ~ 2^(6*n + 3) * 3^(6*n + 3/2) / (31 * sqrt(Pi*n) * 5^(5*n + 1/2)). - Vaclav Kotesovec, Mar 28 2020

#9 by Vaclav Kotesovec at Sat Mar 28 07:10:36 EDT 2020

MATHEMATICA

Join[{1}, Table[4*Binomial[6*n-1, n] * HypergeometricPFQ[{1, -5*n, -n}, {1/2 - 3*n, 1 - 3*n}, 1/4]/5, {n, 1, 20}]] (* Vaclav Kotesovec, Mar 28 2020 *)

STATUS

approved

editing

#8 by Bruno Berselli at Sun Mar 22 17:31:52 EDT 2020

STATUS

proposed

approved