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A328874 Constant term in the expansion of (-1 + (1 + x + 1/x) * (1 + y + 1/y) * (1 + z + 1/z))^n.

%I A328874 #37 Oct 30 2019 08:04:20
%S A328874 1,0,26,264,5646,101520,2103740,43632960,942507790,20685977760,
%T A328874 462661368876,10483696885200,240373512418116,5564581640601984,
%U A328874 129901678525143096,3054381796821779424,72272856926974596750,1719662128611006026304,41120565854695068532076,987633314722818034066224
%N A328874 Constant term in the expansion of (-1 + (1 + x + 1/x) * (1 + y + 1/y) * (1 + z + 1/z))^n.
%C A328874 Also number of n-step closed walks (from origin to origin) in cubic lattice, using steps (t_1,t_2,t_3) (t_k = -1, 1 or 0 for 1 <= k <= 3) except for (0,0,0).
%H A328874 Seiichi Manyama, <a href="/A328874/b328874.txt">Table of n, a(n) for n = 0..710</a>
%F A328874 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^3.
%F A328874 From _Vaclav Kotesovec_, Oct 30 2019: (Start)
%F A328874 Recurrence: (n-1)*n^3*(5103*n^4 - 38556*n^3 + 107838*n^2 - 132564*n + 60401)*a(n) = (n-1)^2*(71442*n^6 - 575505*n^5 + 1817613*n^4 - 2850549*n^3 + 2299999*n^2 - 891084*n + 132528)*a(n-1) + (1505385*n^8 - 17395560*n^7 + 85857516*n^6 - 235935678*n^5 + 393710399*n^4 - 407039414*n^3 + 253464484*n^2 - 86477832*n + 12324048)*a(n-2) + 2*(n-2)*(1224720*n^7 - 13539960*n^6 + 61400268*n^5 - 146649411*n^4 + 197630220*n^3 - 149760433*n^2 + 59083626*n - 9168258)*a(n-3) - 4*(n-3)*(n-2)*(1153278*n^6 - 11020212*n^5 + 40809852*n^4 - 74540514*n^3 + 70559711*n^2 - 32643654*n + 5797748)*a(n-4) - 8*(n-4)*(n-3)*(n-2)*(1367604*n^5 - 9649206*n^4 + 23421096*n^3 - 25438791*n^2 + 12638258*n - 2271566)*a(n-5) - 1040*(n-5)*(n-4)*(n-3)*(n-2)*(5103*n^4 - 18144*n^3 + 22788*n^2 - 12144*n + 2222)*a(n-6).
%F A328874 a(n) ~ 13 * 26^(n + 1/2) / (108 * Pi^(3/2) * n^(3/2)). (End)
%t A328874 Table[Sum[(-1)^(n-k) * Binomial[n, k] * Sum[Binomial[k, 2*j]*Binomial[2*j, j], {j, 0, k}]^3, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 30 2019 *)
%t A328874 Table[Sum[(-1)^(n-k) * Binomial[n, k] * Hypergeometric2F1[1/2 - k/2, -k/2, 1, 4]^3, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 30 2019 *)
%o A328874 (PARI) {a(n) = polcoef(polcoef(polcoef((-1+(1+x+1/x)*(1+y+1/y)*(1+z+1/z))^n, 0), 0), 0)}
%o A328874 (PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^3)}
%Y A328874 Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m: A126869 (m=1), A094061 (m=2), this sequence (m=3), A328875 (m=4).
%Y A328874 Cf. A326920.
%K A328874 nonn
%O A328874 0,3
%A A328874 _Seiichi Manyama_, Oct 29 2019