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Number of ways to color edges of a tetrahedron using <= n colors.
8

%I #30 Feb 29 2024 17:47:36

%S 0,1,12,87,416,1475,4236,10437,22912,45981,85900,151371,254112,409487,

%T 637196,962025,1414656,2032537,2860812,3953311,5373600,7196091,

%U 9507212,12406637,16008576,20443125,25857676,32418387,40311712

%N Number of ways to color edges of a tetrahedron using <= n colors.

%H Vincenzo Librandi, <a href="/A046023/b046023.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = (n^6+3*n^4+8*n^2)/12.

%F G.f.: x*(1+x)*(1+4*x+20*x^2+4*x^3+x^4)/(1-x)^7. - _Colin Barker_, Jan 30 2012

%F E.g.f.: exp(x)*x*(12 + 60*x + 108*x^2 + 68*x^3 + 15*x^4 + x^5)/12. - _Stefano Spezia_, Feb 29 2024

%p A046023 := n->(n^6+3*n^4+8*n^2)/12;

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,12,87,416,1475,4236},30] (* _Vincenzo Librandi_, Jan 31 2012 *)

%o (PARI) a(n)=(n^6+3*n^4+8*n^2)/12 \\ _Charles R Greathouse IV_, Jan 31 2012

%Y Cf. A006008.

%Y Row 3 of A327083.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Apr 11 2001