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 A368708 #30 Sep 15 2024 14:41:54
%S A368708 1,1,3,13,69,417,2763,19609,146793,1146833,9278595,77292261,659973933,
%T A368708 5756169681,51137399979,461691066417,4228199347281,39216540096993,
%U A368708 367890444302787,3486697883136957,33353178454762389,321754445379041601,3127955713554766923,30624486778208481993,301790556354721667769,2991957347531210976817
%N A368708 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -2).
%F A368708 a(n) = (1/2)*B(n, 2) where B(n, x) are the Baxter polynomials with coefficients A359363, for n > 0. - _Peter Luschny_, Jan 04 2024
%F A368708 a(n) ~ 3^(n + 7/6) * (2^(2/3) + 2^(1/3) + 1)^(n + 5/3) / (2^(4/3) * Pi * n^4). - _Vaclav Kotesovec_, Jan 04 2024
%F A368708 a(0) = 1, a(n) = 2^(n + 1)/(n*(n + 1)^2)*Sum_{k=1..n} (1/2)^k*binomial(n + 1, k - 1)*binomial(n + 1, k)*binomial(n + 1, k + 1). - _Detlef Meya_, May 29 2024
%F A368708 From _Peter Bala_, Sep 09 2024: (Start)
%F A368708 a(n+1) = Sum_{k = 0..n} A056939(n, k)*2^k.
%F A368708 P-recursive: (n+1)*(n+3)*(n+2)*(3*n-2)*a(n) = 3*(9*n^3+3*n^2-4*n+4)*(n+1)*a(n-1) + 3*(n-2)*(3*n-1)*(9*n^2-3*n-10)*a(n-2) + 27*(3*n+1)*(n-3)*(n-2)^2*a(n-3) = 0 with a(0) = 1, a(1) = 1 and a(2) = 3. (End)
%p A368708 seq(simplify( hypergeom([-1 - n, -n, 1 - n], [2, 3], -2) ), n = 0..25); # _Peter Bala_, Sep 09 2024
%t A368708 Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -2], {n, 0, 30}] (* _Vaclav Kotesovec_, Jan 04 2024 *)
%t A368708 a[0] := 1; a[n_] := 2^(n + 1)/(n*(n + 1)^2)*Sum[(1/2)^k*Binomial[n + 1, k - 1]*Binomial[n + 1, k]*Binomial[n + 1, k + 1], {k, 1, n}]; Table[a[n], {n, 0, 25}] (* _Detlef Meya_, May 28 2024 *)
%o A368708 (SageMath)
%o A368708 def A368708(n): return PolyA359363(n, 2) // 2 if n > 0 else 1
%o A368708 print([A368708(n) for n in range(23)]) # _Peter Luschny_, Jan 04 2024
%o A368708 (Python)
%o A368708 def A368708(n):
%o A368708 if n == 0: return 1
%o A368708 return sum(2**k * v for k, v in enumerate(A359363Row(n))) // 2
%o A368708 print([A368708(n) for n in range(26)]) # _Peter Luschny_, Jan 04 2024
%Y A368708 Cf. A001181, A007724, A056939, A217800, A359363, A368709.
%K A368708 nonn,easy
%O A368708 0,3
%A A368708 _Joerg Arndt_, Jan 04 2024