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 A361892 #23 Mar 25 2025 17:56:10
%S A361892 1,43,9451,6031627,6571985126,9140730357409,14801600281919487,
%T A361892 26927918031565051915,53804800109969394477580,
%U A361892 116002825041515533807200418,266118189111094898593879923346,642598035707739308769581970619393
%N A361892 a(n) = S(7,2*n-1)/S(1,2*n-1), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.
%C A361892 Odd bisection of A361891.
%C A361892 Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for positive integers n and r and all primes p >= 5.
%H A361892 Seiichi Manyama, <a href="/A361892/b361892.txt">Table of n, a(n) for n = 1..282</a>
%F A361892 a(n) = 1/binomial(2*n-1,n-1) * Sum_{k = 0..n-1} ( (2*n - 2*k)/(2*n - k) * binomial(2*n-1,k) )^7 for n >= 1.
%F A361892 a(n) ~ 3 * 2^(12*n+1) / (2401 * Pi^3 * n^6). - _Vaclav Kotesovec_, Mar 24 2025
%p A361892 seq(add( ( binomial(2*n-1,k) - binomial(2*n-1,k-1) )^7/binomial(2*n-1,n-1), k = 0..n-1), n = 1..20);
%t A361892 Table[Sum[(Binomial[2*n-1, k]-Binomial[2*n-1, k-1])^7 / Binomial[2*n-1, n-1], {k, 0, n-1}], {n, 1, 20}] (* _Vaclav Kotesovec_, Mar 24 2025 *)
%o A361892 (Python)
%o A361892 from math import comb
%o A361892 def A361892(n): return sum((comb((n<<1)-1,j)*(m:=n-j<<1)//(m+j))**7 for j in range(n))//comb((n<<1)-1,n-1) # _Chai Wah Wu_, Mar 25 2025
%Y A361892 Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A382394 ( S(3,2*n-1) ), A183069 ( S(3,2*n-1)/ S(1,2*n-1) ), A361887 ( S(5,n) ), A361888 ( S(5,n)/S(1,n) ), A361889 ( S(5,2*n-1)/S(1,2*n-1) ), A361890 ( S(7,n) ), A361891 ( S(7,n)/S(1,n) ).
%K A361892 nonn,easy
%O A361892 1,2
%A A361892 _Peter Bala_, Mar 30 2023
%E A361892 Offset changed to 1 by _Georg Fischer_, Nov 20 2024