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A328874
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Constant term in the expansion of (-1 + (1 + x + 1/x) * (1 + y + 1/y) * (1 + z + 1/z))^n.
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5
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1, 0, 26, 264, 5646, 101520, 2103740, 43632960, 942507790, 20685977760, 462661368876, 10483696885200, 240373512418116, 5564581640601984, 129901678525143096, 3054381796821779424, 72272856926974596750, 1719662128611006026304, 41120565854695068532076, 987633314722818034066224
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^3.
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).
a(n) ~ 13 * 26^(n + 1/2) / (108 * Pi^(3/2) * n^(3/2)). (End)
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MATHEMATICA
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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 *)
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 *)
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PROG
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(PARI) {a(n) = polcoef(polcoef(polcoef((-1+(1+x+1/x)*(1+y+1/y)*(1+z+1/z))^n, 0), 0), 0)}
(PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^3)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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