A382472 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(2*k,2*n-2*k).
1, 6, 27, 182, 987, 4620, 20678, 87732, 355095, 1387462, 5258967, 19416222, 70086803, 248046540, 862694058, 2954279732, 9977518122, 33278815920, 109749059308, 358231786128, 1158357919194, 3713416860580, 11810098024410, 37285901203740, 116917784689237
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (12,-54,112,-141,312,-674,588,-714,1812,-1134,888,-2741,888,-1134,1812,-714,588,-674,312,-141,112,-54,12,-1).
Programs
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Magma
[&+[Binomial(k+5, 5) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 11 2025
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Mathematica
Table[Sum[Binomial[k+5,5]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 11 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(k+5, 5)*binomial(2*k, 2*n-2*k));
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PARI
my(N=5, 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..3} 4^k * binomial(6,2*k) * (1-x-x^2)^(6-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^6.