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

0,2

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).

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.

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 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(k+6, 6)*binomial(2*k, 2*n-2*k));
(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))
(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

Crossrefs

Cf. A108479, A381421, A382230, A382470, A382471, A382472, A382474.

Cf. A034839, A377158.

Sequence in context: A177261 A105926 A184460 * A096686 A165505 A165639

Adjacent sequences: A382470 A382471 A382472 * A382474 A382475 A382476

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 28 2025

Status

approved