A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,2*n-2*k).
1, 3, 9, 46, 171, 591, 2033, 6714, 21606, 68308, 212370, 651234, 1974113, 5924277, 17623671, 52025858, 152539077, 444530073, 1288396257, 3715833732, 10668907932, 30507914696, 86912853588, 246755125332, 698353551105, 1970673504951, 5545952371509, 15568330002486
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (6,-9,2,-18,30,7,30,-18,2,-9,6,-1).
Programs
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Magma
[&+[Binomial(k+2, 2) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 22 2025
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Mathematica
Table[Sum[Binomial[k+2,2]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 22 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(2*k, 2*n-2*k));
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PARI
my(N=2, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); 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.: (Sum_{k=0..1} 4^k * binomial(3,2*k) * (1-x-x^2)^(3-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^3.
a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) - 18*a(n-4) + 30*a(n-5) + 7*a(n-6) + 30*a(n-7) - 18*a(n-8) + 2*a(n-9) - 9*a(n-10) + 6*a(n-11) - a(n-12).