OEIS/Coordination sequences for lattices

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Revision as of 13:50, 28 July 2020 by imported>Gfis (Created page with "The recurrences were derived by Mathematica: bin[n_,k_]:=(n!/(n-k)!/k!); === C_n === * [https://oeis.org/A103884 A103884 (C_4)] d:= 4; CoefficientList[Series[(Sum[Binomial[...")
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The recurrences were derived by Mathematica: 

bin[n_,k_]:=(n!/(n-k)!/k!);

C_n

d:= 4; CoefficientList[Series[(Sum[Binomial[2n,2k]*x^k, {k, 0, d}])/(1-x)^d, {x,0,11}],x]
{1, 32, 192, 608, 1408, 2720, 4672, 7392, 11008, 15648, 21440, 28512}
InputForm[FullSimplify[bin[2*n,2*k]/bin[2*n,2*k-2]]] -> ((-1 + 2*k - 2*n)*(-1 + k - n))/(k*(-1 + 2*k))
n:=4; f:= gfun:-rectoproc({a(k)=(2*n-2*k+1)*(n-(k-1))/(k*(2*k-1))*a(k-1),a(0)=1},a(k),remember): map(f, [$0
  Recurrence: a(k)=(2*n-2*k+1)*(n-(k-1))/(k*(2*k-1))*a(k-1), a(0)=1, k=0..n
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