A382842 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^3.
1, 1, 9, 217, 1945, 35001, 764001, 12079089, 250222617, 5424133465, 107360983009, 2358751625649, 52540471866961, 1147794435985393, 26151265459123065, 600227875293254217, 13779170435209475097, 322302377797126709913, 7582484532013652243169, 179184911648568670363185, 4275721755296040840336945
Offset: 0
Keywords
Programs
-
Maple
a:= n-> add(combinat[multinomial](n, n-2*k, k$2)^3, k=0..n/2): seq(a(n), n=0..20); # Alois P. Heinz, Apr 07 2025
-
Mathematica
Table[Sum[(Binomial[n, k] Binomial[n - k, k])^3, {k, 0, Floor[n/2]}], {n, 0, 20}] Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, 1/2 - n/2, -n/2, -n/2, -n/2}, {1, 1, 1, 1, 1}, 64], {n, 0, 20}] Table[SeriesCoefficient[1/((1 - x) (1 - y) (1 - z) (1 - u) (1 - v) (1 - w) - (x y z)^2 u v w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {u, 0, n}, {v, 0, n}, {w, 0, n}], {n, 0, 20}]
Formula
a(n) ~ 3^(3*n+3) / (8 * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Apr 07 2025
a(n) = Sum_{k=0..floor(n/2)} A089627(n,k)^3. - Alois P. Heinz, Apr 07 2025
Comments