A382473 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(2*k,2*n-2*k).
1, 7, 35, 252, 1498, 7602, 36498, 165600, 713769, 2957647, 11850223, 46111352, 174956250, 649284286, 2362771938, 8449241836, 29744151416, 103237104740, 353744829032, 1198001464940, 4013905507150, 13316690882670, 43780154987030, 142726581203640
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (14, -77, 210, -336, 602, -1435, 2018, -1981, 4312, -5894, 3360, -7721, 9562, -3079, 9562, -7721, 3360, -5894, 4312, -1981, 2018, -1435, 602, -336, 210, -77, 14, -1).
Programs
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Magma
[&+[Binomial(k+6, 6) * 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+6,6]*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+6, 6)*binomial(2*k, 2*n-2*k));
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PARI
my(N=6, 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(7,2*k) * (1-x-x^2)^(7-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^7.