A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2*k,2*n-2*k).
1, 2, 5, 22, 68, 206, 631, 1870, 5467, 15836, 45416, 129260, 365565, 1028122, 2877697, 8021010, 22274476, 61653850, 170152275, 468347046, 1286055927, 3523777912, 9635982160, 26302324504, 71674754873, 195015074610, 529846108989, 1437657038030, 3896050721940
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (4,-2,0,-11,0,-2,4,-1).
Programs
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Magma
[&+[(k+1) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 23 2025
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Mathematica
Table[Sum[(k+1)*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 23 2025 *) -
PARI
a(n) = sum(k=0, n, (k+1)*binomial(2*k, 2*n-2*k));
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PARI
my(N=1, 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.: ((1-x-x^2)^2 + 4*x^3) / ((1-x-x^2)^2 - 4*x^3)^2.
a(n) = 4*a(n-1) - 2*a(n-2) - 11*a(n-4) - 2*a(n-6) + 4*a(n-7) - a(n-8).