A117202 - OEIS

%I #76 Jul 01 2026 18:31:39

%S 0,1,4,15,52,170,534,1631,4880,14373,41810,120406,343884,975325,

%T 2749852,7713435,21540304,59917826,166094370,458998523,1264919720,

%U 3477182961,9536877614,26102772910,71309161752,194468551225,529490287924,1439539861671,3908348148460,10597623588698

%N Binomial transform of n*F(n).

%C Binomial transform of A045925.

%C Number of acyclic subgraphs of the wheel graph W_n (on n+1 vertices) with exactly n-1 edges. - _Emil R. Vaughan_, Jun 12 2007

%C Equivalently, number of two-component spanning forests of the wheel graph W_n (on n+1 vertices). - _Harry Richman_, Jul 31 2023

%C Starting (1, 4, 15, 52, ...) = binomial transform of A136376. - _Gary W. Adamson_, Sep 03 2008

%H Indranil Ghosh, <a href="/A117202/b117202.txt">Table of n, a(n) for n = 0..1000</a>

%H Harry Richman, Farbod Shokrieh, and Chenxi Wu, <a href="https://arxiv.org/abs/2308.03859">Counting two-forests and random cut size via potential theory</a>, arXiv:2308.03859 [math.CO], 2023. See p. 18.

%H Jesús Salas and Alan D. Sokal, <a href="https://doi.org/10.1007/s10955-009-9725-1">Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial</a>, J. Stat. Phys. 135 (2009), 279-373; <a href="https://arxiv.org/abs/0711.1738">arXiv preprint</a>, arXiv:0711.1738 [cond-mat.stat-mech], 2007-2009. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).

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

%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).

%F a(n) = Sum_{k=0..n} C(n,k)*k*F(k).

%F From _Benoit Cloitre_, Nov 29 2006: (Start)

%F a(n) = Sum_{k=1..n} F(2k)*B(2n-2k)*binomial(2n,2k) where F=Fibonacci numbers and B=Bernoulli numbers.

%F a(n) = n*F(2n-1). (End)

%F a(n) = (2^(-1-n)*(-(-5+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(5+sqrt(5)))*n) / 5. - _Colin Barker_, Feb 26 2017

%F From _Harry Richman_, Jul 31 2023: (Start)

%F a(n) = (1/sqrt(5)) * n * (((1 + sqrt(5)) / 2)^(2*n-1) - ((1 - sqrt(5)) / 2)^(2*n-1)).

%F a(n) = round((1/sqrt(5)) * n * phi^(2n-1)), where phi = (1+sqrt(5))/2 is the golden ratio A001622. (End)

%F E.g.f.: x * exp(3*x/2) * (cosh(sqrt(5)*x/2) + sinh(sqrt(5)*x/2)/sqrt(5)). - _Amiram Eldar_, Jan 24 2026

%t Table[n Fibonacci[2n-1],{n,0,26}] (* _Indranil Ghosh_, Feb 26 2017 *)

%t (* Alternative: *)

%t Table[Sum[Fibonacci[2k]*BernoulliB[2n-2k]*Binomial[2n,2k],{k,1,n}],{n,0,26}] (* _Indranil Ghosh_, Feb 26 2017 *)

%t (* Alternative: *)

%t CoefficientList[Series[x(1-2x+2x^2)/(1-3x+x^2)^2 ,{x,0,26}],x] (* _Indranil Ghosh_, Feb 26 2017 *)

%o (PARI) a(n) = n*fibonacci(2*n-1); \\ _Indranil Ghosh_, Feb 26 2017

%o (PARI) concat(0, Vec(x*(1-2*x+2*x^2) / (1-3*x+x^2)^2 + O(x^30))) \\ _Colin Barker_, Feb 26 2017

%Y Cf. A000045, A001519, A001622, A045925, A111262, A136376.

%Y Cf. A004146 (number of spanning trees of wheel graph).

%K easy,nonn,changed

%O 0,3

%A _Paul Barry_, Mar 02 2006