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 A361677 #10 Mar 22 2023 06:36:11 %S A361677 1,1,1,19,73,181,1711,10081,38809,256033,1696861,8388271,49449511, %T A361677 326195299,1847392093,10789655059,69202030969,418647580489, %U A361677 2498113460881,15735859252147,97919649290053,598317173139313,3748943081117323 %N A361677 Constant term in the expansion of (1 + x + y + z + 1/(x*y) + 1/(y*z) + 1/(z*x))^n. %F A361677 a(n) = Sum_{k=0..floor(n/3)} (3*k)!/k!^3 * binomial(3*k,k) * binomial(n,3*k). %F A361677 From _Vaclav Kotesovec_, Mar 22 2023: (Start) %F A361677 Recurrence: 2*n^3*(2*n - 3)*a(n) = 2*(10*n^4 - 32*n^3 + 38*n^2 - 22*n + 5)*a(n-1) - 2*(n-1)*(2*n - 3)*(10*n^2 - 24*n + 17)*a(n-2) + (n-2)*(n-1)*(769*n^2 - 2331*n + 1594)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*(739*n - 1481)*a(n-4) + 733*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5). %F A361677 a(n) ~ sqrt(733/108 + 1/2^(2/3) + 9/2^(4/3)) * (1 + 9/2^(2/3))^n / (2 * Pi^(3/2) * n^(3/2)). (End) %t A361677 Table[Sum[(3*k)!/k!^3 * Binomial[3*k,k] * Binomial[n,3*k], {k,0,n/3}], {n,0,25}] (* _Vaclav Kotesovec_, Mar 22 2023 *) %o A361677 (PARI) a(n) = sum(k=0, n\3, (3*k)!/k!^3*binomial(3*k, k)*binomial(n, 3*k)); %Y A361677 Cf. A201805, A361678. %Y A361677 Cf. A275047, A344560. %K A361677 nonn %O A361677 0,4 %A A361677 _Seiichi Manyama_, Mar 20 2023