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 A361673 #18 Mar 22 2023 07:55:13 %S A361673 1,1,1,1,1,61,361,1261,3361,7561,34021,235621,1294921,5482621, %T A361673 19039021,65345281,286147681,1511480881,7688794681,34337600281, %U A361673 138221512741,554603041441,2454508134541,11874549049441,57412094595241,261925516443361,1134301869703861 %N A361673 Constant term in the expansion of (1 + x*y + y*z + z*x + 1/(x*y*z))^n. %C A361673 Also constant term in the expansion of (1 + x^2 + y^2 + z^2 + 1/(x*y*z))^n. %H A361673 Seiichi Manyama, <a href="/A361673/b361673.txt">Table of n, a(n) for n = 0..1000</a> %F A361673 a(n) = n! * Sum_{k=0..floor(n/5)} 1/(k!^3 * (2*k)! * (n-5*k)!) = Sum_{k=0..floor(n/5)} binomial(n,5*k) * A001460(k). %F A361673 From _Vaclav Kotesovec_, Mar 22 2023: (Start) %F A361673 Recurrence: 2*n^3*(2*n - 5)*a(n) = 2*(10*n^4 - 40*n^3 + 50*n^2 - 30*n + 7)*a(n-1) - 10*(n-1)*(4*n^3 - 18*n^2 + 26*n - 13)*a(n-2) + 40*(n-2)^3*(n-1)*a(n-3) - 10*(n-3)*(n-2)*(n-1)*(2*n - 5)*a(n-4) + 3129*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5). %F A361673 a(n) ~ sqrt(c) * (1 + 5/2^(2/5))^n / (Pi^(3/2) * n^(3/2)), where c = 3.154712586460560795509193778252140601572145506226776094640234924884123818... is the real root of the equation -30634915689 + 95407210000*c - 127160000000*c^2 + 79846400000*c^3 - 25600000000*c^4 + 3276800000*c^5 = 0. (End) %t A361673 Table[n! * Sum[1/(k!^3 * (2*k)! * (n-5*k)!), {k,0,n/5}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 22 2023 *) %o A361673 (PARI) a(n) = n!*sum(k=0, n\5, 1/(k!^3*(2*k)!*(n-5*k)!)); %Y A361673 Cf. A344560, A361675. %Y A361673 Cf. A361637, A361658. %Y A361673 Cf. A001460. %K A361673 nonn %O A361673 0,6 %A A361673 _Seiichi Manyama_, Mar 20 2023