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%I #15 Mar 22 2023 07:54:33
%S 1,1,1,1,1,1,1,211,1681,7561,25201,69301,166321,360361,990991,5405401,
%T 34834801,187867681,833709241,3153281041,10491944401,31945216801,
%U 97323704941,345845431471,1529597398561,7451402805001,35092646589001,151591791651301
%N Constant term in the expansion of (1 + x^4 + y^4 + z^4 + 1/(x*y*z))^n.
%H Winston de Greef, <a href="/A361701/b361701.txt">Table of n, a(n) for n = 0..1611</a>
%F a(n) = Sum_{k=0..floor(n/7)} (3*k)!/k!^3 * binomial(7*k,3*k) * binomial(n,7*k).
%F From _Vaclav Kotesovec_, Mar 22 2023: (Start)
%F Recurrence: 8*n^3*(2*n - 7)*(4*n - 21)*(4*n - 7)*a(n) = 8*(224*n^6 - 2688*n^5 + 11550*n^4 - 22736*n^3 + 22666*n^2 - 11746*n + 2475)*a(n-1) - 56*(n-1)*(96*n^5 - 1200*n^4 + 5540*n^3 - 11982*n^2 + 12466*n - 5115)*a(n-2) + 224*(n-2)*(n-1)*(40*n^4 - 480*n^3 + 2065*n^2 - 3822*n + 2607)*a(n-3) - 56*(n-3)*(n-2)*(n-1)*(160*n^3 - 1680*n^2 + 5730*n - 6407)*a(n-4) + 112*(n-4)*(n-3)*(n-2)*(n-1)*(48*n^2 - 384*n + 757)*a(n-5) - 896*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 9)*a(n-6) + 823799*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).
%F a(n) ~ sqrt(c) * (1 + 7/2^(8/7))^n / (Pi^(3/2) * n^(3/2)), where c = 3.4855654710461411310762468259332410505173151761420224383969482891017005063... is the real root of the equation -559066901335151399 + 2527163634923732000*c - 5081793740448746496*c^2 + 5406293137205395456*c^3 - 3558495001867452416*c^4 + 1393309590535274496*c^5 - 303305489096114176*c^6 + 28296722014797824*c^7 = 0. (End)
%t Table[Sum[(3*k)!/k!^3 * Binomial[7*k,3*k] * Binomial[n,7*k], {k,0,n/7}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 22 2023 *)
%o (PARI) a(n) = sum(k=0, n\7, (3*k)!/k!^3*binomial(7*k, 3*k)*binomial(n, 7*k));
%Y Cf. A361637, A361658, A361673.
%Y Cf. A361677.
%K nonn,easy
%O 0,8
%A _Seiichi Manyama_, Mar 21 2023
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