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Number of (undirected) paths in the n-helm graph.
0

%I #9 Sep 09 2017 11:02:26

%S 3,26,93,260,615,1278,2401,4168,6795,10530,15653,22476,31343,42630,

%T 56745,74128,95251,120618,150765,186260,227703,275726,330993,394200,

%U 466075,547378,638901,741468,855935,983190,1124153,1279776,1451043,1638970,1844605,2069028

%N Number of (undirected) paths in the n-helm graph.

%C Extended to a(1)-a(2) using the formula/recurrence.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).

%F a(n) = n*(-21 + 38*n - 12*n^2 + 4*n^3)/3.

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

%F G.f.: x*(-3 - 11*x + 7*x^2 - 25*x^3)/(-1 + x)^5.

%t Table[n (-21 + 38 n - 12 n^2 + 4 n^3)/3, {n, 20}]

%t LinearRecurrence[{5, -10, 10, -5, 1}, {3, 26, 93, 260, 615}, 20]

%t CoefficientList[Series[(-3 - 11 x + 7 x^2 - 25 x^3)/(-1 + x)^5, {x, 0, 20}], x]

%K nonn,easy

%O 1,1

%A _Eric W. Weisstein_, Sep 07 2017