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 A361885 #18 Mar 30 2023 05:09:01
%S A361885 9,979,165816,33372819,7380882509,1732912534168,424032181044264,
%T A361885 106952563532680339,27609695174536836075,7259294757681340436979,
%U A361885 1937215339689731617386000,523352118643145676922317336,142854011885066484369862826496,39337931825265398967484384872560
%N A361885 a(n) = (1/n) * Sum_{k = 0..2*n} (n+2*k) * binomial(n+k-1,k)^3.
%C A361885 Compare with the closed form evaluation of the binomial sums (1/n) * Sum_{k = 0..2*n} (-1)^k * (n + 2*k) * binomial(n+k-1,k) = binomial(3*n,n) and (1/n) * Sum_{k = 0..2*n} (n + 2*k) * binomial(n+k-1,k)^2 = binomial(3*n,n)^2.
%C A361885 The binomial coefficients u(n) := binomial(3*n,n) = A005809(n) satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for positive integers n and r and all primes p >= 5. We conjecture that the present sequence satisfies the same congruences.
%C A361885 More generally, for m >= 3, the sequences {b_m(n) : n >= 1} and {c_m(n) : n >= 1} defined by b_m(n) = (1/n) * Sum_{k = 0..2*n} (n + 2*k) * binomial(n+k-1,k)^m and c_m(n) = (1/n) * Sum_{k = 0..2*n} (-1)^k * (n + 2*k) * binomial(n+k-1,k)^m may satisfy the same congruences.
%F A361885 a(n) ~ 5 * 3^(9*n + 3/2) / (19 * Pi^(3/2) * n^(3/2) * 2^(6*n + 3)). - _Vaclav Kotesovec_, Mar 29 2023
%p A361885 seq( (1/n)*add((n + 2*k) * binomial(n+k-1,k)^3, k = 0..2*n), n = 1..20);
%t A361885 Table[Sum[(n+2*k) * Binomial[n+k-1,k]^3, {k,0,2*n}]/n, {n,1,20}] (* _Vaclav Kotesovec_, Mar 29 2023 *)
%o A361885 (PARI) a(n) = (1/n) * sum(k = 0, 2*n, (n+2*k) * binomial(n+k-1,k)^3); \\ _Michel Marcus_, Mar 30 2023
%Y A361885 Cf. A005809, A188662, A361883, A361884, A361886.
%K A361885 nonn,easy
%O A361885 0,1
%A A361885 _Peter Bala_, Mar 28 2023