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%I #12 Mar 22 2023 06:52:12
%S 1,1,1,1,97,481,1441,3361,77281,647137,3195361,11674081,116286721,
%T 1147935361,7611379777,37451144641,263670781921,2456043418081,
%U 19073086806241,115319128034017,748239468100417,6179458007222977,50636218964639617,350400618132423937
%N Constant term in the expansion of (1 + w + x + y + z + 1/(x*y*z) + 1/(w*y*z) + 1/(w*x*z) + 1/(w*x*y))^n.
%F a(n) = Sum_{k=0..floor(n/4)} (4*k)!/k!^4 * binomial(4*k,k) * binomial(n,4*k).
%F From _Vaclav Kotesovec_, Mar 22 2023: (Start)
%F Recurrence: 3*n^4*(3*n - 8)*(3*n - 4)*a(n) = 3*(63*n^6 - 405*n^5 + 1015*n^4 - 1355*n^3 + 1049*n^2 - 439*n + 77)*a(n-1) - 3*(n-1)*(189*n^5 - 1485*n^4 + 4685*n^3 - 7575*n^2 + 6313*n - 2163)*a(n-2) + 3*(n-2)*(n-1)*(315*n^4 - 2610*n^3 + 8285*n^2 - 12030*n + 6749)*a(n-3) + (n-3)*(n-2)*(n-1)*(64591*n^3 - 385926*n^2 + 701651*n - 375786)*a(n-4) - 3*(n-4)*(n-3)*(n-2)*(n-1)*(65347*n^2 - 326519*n + 391384)*a(n-5) + 3*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(65473*n - 196383)*a(n-6) - 65509*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).
%F a(n) ~ sqrt(65563/12288 + 3*sqrt(3)/8 + sqrt(3/8 + 65563/(4096*sqrt(3)))) * (1 + 16/3^(3/4))^n / (Pi^2 * n^2). (End)
%t Table[Sum[(4*k)!/k!^4 * Binomial[4*k,k] * Binomial[n,4*k], {k,0,n/4}], {n,0,25}] (* _Vaclav Kotesovec_, Mar 22 2023 *)
%o (PARI) a(n) = sum(k=0, n\4, (4*k)!/k!^4*binomial(4*k, k)*binomial(n, 4*k));
%Y Cf. A201805, A361677.
%Y Cf. A361637.
%K nonn
%O 0,5
%A _Seiichi Manyama_, Mar 20 2023
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