This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357955 #47 Mar 23 2025 20:51:56
%S A357955 37,60,108,60,-660,60,82404,1411848,17540460,191318820,1952058108,
%T A357955 19175376324,184118073828,1743153802320,16359157606200,
%U A357955 152693295412560,1420516291306860,13190159377278324,122358232382484420,1134645084249344400,10522118980232969340
%N A357955 a(n) = 3*binomial(4*n,n) - 20*binomial(3*n,n) + 54*binomial(2*n,n).
%C A357955 Conjectures:
%C A357955 1) a(p) == a(1) (mod p^7) for all primes p >= 3 except p = 7.
%C A357955 2) a(p^r) == a(p^(r-1)) ( mod p^(3*r+5) ) for r >= 2 and all primes p >= 3.
%C A357955 These are stronger supercongruences than those satisfied separately by the sequences {binomial(2*n,n)} = A000984, {binomial(3*n,n)} = A005809 and {binomial(4*n,n)} = A005810.
%C A357955 Conjecture 1) was proved by Aidagulov and Alekseyev; see the remarks following Corollary 2. - _Peter Bala_, Oct 29 2022
%H A357955 Paolo Xausa, <a href="/A357955/b357955.txt">Table of n, a(n) for n = 0..1000</a>
%H A357955 R. R. Aidagulov and M. A. Alekseyev, <a href="https://doi.org/10.1007/s10958-018-3948-0">On p-adic approximation of sums of binomial coefficients</a>, Journal of Mathematical Sciences 233:5 (2018), 626-634; <a href="https://arxiv.org/abs/1602.02632">arXiv:1602.02632 [math.NT]</a>, 2018.
%H A357955 C. Helou and G. Terjanian, <a href="https://doi.org/10.1016/j.jnt.2007.06.008">On Wolstenholme’s theorem and its converse</a>, J. Number Theory 128 (2008), 475-499.
%H A357955 Romeo Meštrović, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv preprint arXiv:1111.3057 [math.NT], 2011.
%F A357955 a(n) = 3*A005810(n) - 20*A005809(n) + 54*A000984(n).
%F A357955 a(k*p^r) == a(k*p^(r-1)) ( mod p^(3*r) ) for positive integers k and r and for all primes p >= 5 (see Meštrović, Section 6, equation 39).
%F A357955 a(p) == a(1) (mod p^6) for all primes p >= 7 (apply Helou and Terjanian, Section 3, Proposition 2).
%F A357955 From _Stefano Spezia_, Jul 17 2024: (Start)
%F A357955 G.f.: 54/sqrt(1-4*x) - 40*cos(arccos(1-27*x/2)/6)/sqrt(4-27*x) + 3*hypergeom([1/4, 1/2, 3/4], [1/3, 2/3], 4^4*x/3^3).
%F A357955 E.g.f.: 54*exp(2*x)*BesselI(0, 2*x) - 20*hypergeom([1/3, 2/3], [1/2, 1], 27*x/4) + 3*hypergeom([1/4, 1/2, 3/4], [1/3, 2/3, 1], 4^4*x/3^3).
%F A357955 a(n) ~ exp(2*n*arctanh(229/283))*sqrt(6/(n*Pi)). (End)
%e A357955 Examples of supercongruences:
%e A357955 a(11) - a(1) = 19175376324 - 60 = (2^3)*3*(11^7)*41 == 0 (mod 11^7).
%e A357955 a(5^2) - a(5) = 726506045044361132812560 - 60 = (2^2)*3*(5^11)*41*30241552444123 == 0 (mod 5^11).
%p A357955 seq(3*binomial(4*n,n) - 20*binomial(3*n,n) + 54*binomial(2*n,n), n = 0..20);
%t A357955 A357955[n_] := 3*Binomial[4*n, n] - 20*Binomial[3*n, n] + 54*Binomial[2*n, n];
%t A357955 Array[A357955, 25, 0] (* _Paolo Xausa_, Jul 17 2024 *)
%o A357955 (Python)
%o A357955 from math import comb
%o A357955 def A357955(n): return 54*comb(m:=n<<1,n)+3*comb(m<<1,n)-20*comb(m+n,n) # _Chai Wah Wu_, Oct 24 2022
%Y A357955 Cf. A000984, A005809, A005810, A268590, A357509, A357567, A357568, A357569.
%K A357955 sign,easy
%O A357955 0,1
%A A357955 _Peter Bala_, Oct 22 2022