A382471 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(2*k,2*n-2*k).
1, 5, 20, 125, 610, 2611, 10815, 42610, 161005, 590155, 2106362, 7348265, 25141430, 84569395, 280246795, 916465742, 2961805180, 9470735650, 29994694130, 94172180660, 293326457342, 907028460410, 2786036875580, 8505001839950, 25815678641935, 77945771624609
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
- Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-55,172,-250,100,-365,510,-29,510,-365,100,-250,172,-55,50,-35,10,-1).
Programs
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Magma
[&+[Binomial(k+4,4) * Binomial(2*k,2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 10 2025
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Mathematica
Table[Sum[Binomial[k+4,4]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 10 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(k+4, 4)*binomial(2*k, 2*n-2*k));
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PARI
my(N=4, 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..2} 4^k * binomial(5,2*k) * (1-x-x^2)^(5-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^5.