A382495 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(2*k,2*n-4*k).
1, 0, 4, 4, 10, 60, 30, 300, 335, 1000, 2506, 3500, 11879, 17304, 44220, 88592, 161865, 385704, 660964, 1475100, 2807956, 5459860, 11313094, 20816004, 42774780, 80798128, 157292750, 307887904, 579776799, 1138007940, 2146348214, 4126143900, 7878910238, 14878269368
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1500
- Index entries for linear recurrences with constant coefficients, signature (0,8,8,-28,-40,28,72,2,16,20,-80,-114,56,-68,0,35,40,-96,128,-110,64,-28,8,-1).
Programs
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Magma
[&+[Binomial(k+3, 3)*Binomial(2*k, 2*n-4*k): k in [0..n]]: n in [0..40]]; // Vincenzo Librandi, May 12 2025
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Mathematica
Table[Sum[Binomial[k+3,3]*Binomial[2*k, 2*n-4*k],{k,0,Floor[n/2]}],{n,0,33}] (* Vincenzo Librandi, May 12 2025 *) -
PARI
a(n) = sum(k=0, n\2, binomial(k+3, 3)*binomial(2*k, 2*n-4*k));
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PARI
my(N=3, 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.: (Sum_{k=0..2} 4^k * binomial(4,2*k) * (1-x^2-x^3)^(4-2*k) * x^(5*k)) / ((1-x^2-x^3)^2 - 4*x^5)^4.
a(n) = 8*a(n-2) + 8*a(n-3) - 28*a(n-4) - 40*a(n-5) + 28*a(n-6) + 72*a(n-7) + 2*a(n-8) + 16*a(n-9) + 20*a(n-10) - 80*a(n-11) - 114*a(n-12) + 56*a(n-13) - 68*a(n-14) + 35*a(n-16) + 40*a(n-17) - 96*a(n-18) + 128*a(n-19) - 110*a(n-20) + 64*a(n-21) - 28*a(n-22) + 8*a(n-23) - a(n-24).