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A002985 Number of trees in an n-node wheel.

%I A002985 M0783 #28 Feb 16 2025 08:32:27
%S A002985 1,1,1,2,3,6,11,20,36,64,108,179,292,464,727,1124,1714,2585,3866,5724,
%T A002985 8418,12290,17830,25713,36898,52664,74837,105873,149178,209364,292793,
%U A002985 407990,566668,784521,1082848,1490197,2045093,2798895,3820629,5202085
%N A002985 Number of trees in an n-node wheel.
%C A002985 This is the number of nonequivalent spanning trees of the n-wheel graph up to isomorphism of the trees.
%D A002985 F. Harary, P. E. O'Neil, R. C. Read and A. J. Schwenk, The number of trees in a wheel, in D. J. A. Welsh and D. R. Woodall, editors, Combinatorics. Institute of Mathematics and Its Applications. Southend-on-Sea, England, 1972, pp. 155-163.
%D A002985 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002985 Andrew Howroyd, <a href="/A002985/b002985.txt">Table of n, a(n) for n = 1..200</a>
%H A002985 Andrew Howroyd, <a href="/A002985/a002985.txt">Derivation of formula</a>.
%H A002985 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WheelGraph.html">Wheel Graph</a>.
%H A002985 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A002985 a(n) = A003293(n-1) - A008804(n-3). - _Andrew Howroyd_, Oct 09 2017
%e A002985 All trees that span a wheel on 5 nodes are equivalent to one of the following:
%e A002985       o         o         o
%e A002985       |         | \     /   \
%e A002985    o--o--o   o--o  o   o--o  o
%e A002985       |         |           /
%e A002985       o         o         o
%t A002985 terms = 40;
%t A002985 A003293[n_] := SeriesCoefficient[Product[(1-x^k)^(-Ceiling[k/2]), {k, 1, terms}], {x, 0, n}];
%t A002985 A008804[n_] := SeriesCoefficient[1/((1-x)^4 (1+x)^2 (1+x^2)), {x, 0, n}];
%t A002985 a[n_] := A003293[n-1] - A008804[n-3];
%t A002985 Array[a, terms] (* _Jean-François Alcover_, Sep 02 2019 *)
%o A002985 (PARI) \\ here b(n) is A003293 and d(n) is A008804.
%o A002985 b(n)={polcoeff( prod(k=1, n, (1-x^k+x*O(x^n))^-ceil(k/2)), n)}
%o A002985 d(n)={(84+12*(-1)^n+6*I*((-I)^n-I^n)+(85+3*(-1)^n)*n+24*n^2+2*n^3)/96}
%o A002985 a(n)=b(n-1)-d(n-3); \\ _Andrew Howroyd_, Oct 09 2017
%Y A002985 Cf. A003293, A004146, A008804.
%K A002985 nonn
%O A002985 1,4
%A A002985 _N. J. A. Sloane_
%E A002985 Terms a(32) and beyond from _Andrew Howroyd_, Oct 09 2017