A382300 a(n) = Sum_{k=0..floor(n/2)} (k+1) * binomial(2*k,2*n-4*k).
1, 0, 2, 2, 3, 18, 7, 60, 65, 144, 356, 410, 1272, 1722, 3743, 7202, 11482, 25566, 40421, 81610, 147169, 259810, 507267, 867792, 1659112, 2961860, 5362592, 9940420, 17583485, 32564548, 58228386, 105606458, 191831767, 343313042, 625086891, 1119760040, 2023087045
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1500
- Index entries for linear recurrences with constant coefficients, signature (0,4,4,-6,-4,-2,-4,-5,8,-6,4,-1).
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 37); Coefficients(R!( ((1-x^2-x^3)^2 + 4*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^2)); // Vincenzo Librandi, May 11 2025 -
Mathematica
Table[Sum[(k+1)*Binomial[2*k,2*n-4*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *) -
PARI
a(n) = sum(k=0, n\2, (k+1)*binomial(2*k, 2*n-4*k));
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PARI
my(N=1, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))
Formula
G.f.: ((1-x^2-x^3)^2 + 4*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^2.
a(n) = 4*a(n-2) + 4*a(n-3) - 6*a(n-4) - 4*a(n-5) - 2*a(n-6) - 4*a(n-7) - 5*a(n-8) + 8*a(n-9) - 6*a(n-10) + 4*a(n-11) - a(n-12).