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 A328875 Constant term in the expansion of (-1 + (1 + w + 1/w) * (1 + x + 1/x) * (1 + y + 1/y) * (1 + z + 1/z))^n. 5
 1, 0, 80, 2160, 121200, 6136800, 356570960, 21225304800, 1321586558320, 84398804078400, 5518934916677280, 367489108030524480, 24852668879410144080, 1702677155195779963200, 117960677109321028039200, 8251450286371615261498560, 582087494621171173360817520 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also number of n-step closed walks (from origin to origin) in 4-dimensional lattice, using steps (t_1,t_2,t_3,t_4) (t_k = -1, 1 or 0 for 1 <= k <= 4) except for (0,0,0,0). For fixed m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m ~ (3^m - 1)^(n + m/2) / (2^m * 3^(m*(m-1)/2) * Pi^(m/2) * n^(m/2)). - Vaclav Kotesovec, Oct 30 2019 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..528 FORMULA a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^4. a(n) ~ 5 * 80^(n+1) / (729 * Pi^2 * n^2). - Vaclav Kotesovec, Oct 30 2019 MATHEMATICA Table[Sum[(-1)^(n-k) * Binomial[n, k] * Sum[Binomial[k, 2*j]*Binomial[2*j, j], {j, 0, k}]^4, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 30 2019 *) PROG (PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^4)} CROSSREFS Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m: A126869 (m=1), A094061 (m=2), A328874 (m=3), this sequence (m=4). Cf. A326920. Sequence in context: A333546 A132466 A277764 * A154307 A233950 A324071 Adjacent sequences: A328872 A328873 A328874 * A328876 A328877 A328878 KEYWORD nonn AUTHOR Seiichi Manyama, Oct 29 2019 STATUS approved

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Last modified January 31 01:17 EST 2023. Contains 359947 sequences. (Running on oeis4.)