A391724 a(n) = Sum_{k=0..floor(n/4)} 2^k * 3^(n-3*k) * binomial(n-3*k+6,6) * binomial(2*(n-3*k),2*k).

1, 21, 252, 2268, 17052, 115290, 741636, 4705452, 29807631, 187564167, 1160653536, 7015318632, 41344513938, 237976030404, 1341887545224, 7435262750328, 40583278223901, 218600645068953, 1163518783578900, 6125571713285892, 31925464081249032, 164844714547081134

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (42, -819, 9828, -80997, 483714, -2148363, 7156512, -17834859, 33424650, -51882201, 93493764, -225744435, 488827710, -743683761, 894004776, -1583880480, 3641548428, -5618158056, 5531570352, -8632853964, 20814851232, -27062392896, 17978399712, -40158323856, 98945128800, -74134995984, 26207048448, -209850803136, 312412355136, 23808556800, 199145350656, -892547522496, 81901435392, 305701869312, 1878925953024, -622655486208, -1580676539904, -3232567118592, 1333279180800, 3082202855424, 4511562789888, -1439941515264, -3519857037312, -4770599887872, 822823723008, 2468471169024, 3565569466368, -198087192576, -1005673439232, -1782784733184, 0, 182849716224, 548549148672, 0, 0, -78364164096).

Formula

G.f.: (Sum_{k=0..3} 72^k * binomial(7,2*k) * (1-3*x-6*x^4)^(7-2*k) * x^(5*k)) / ((1-3*x-6*x^4)^2 - 72*x^5)^7.

Mathematica

Table[Sum[2^k*3^(n-3*k)*Binomial[n-3*k+6, 6]*Binomial[2*(n-3*k), 2*k], {k, 0, Floor[n/4]}], {n, 0, 30}] (* Vincenzo Librandi, Dec 21 2025 *)

Prog

(PARI) my(A=2, B=3, C=4*A*B^2, N=7, M=30, x='x+O('x^M), X=1-B*x-A*B*x^4, Y=5); Vec(sum(k=0, N\2, C^k*binomial(N, 2*k)*X^(N-2*k)*x^(Y*k))/(X^2-C*x^Y)^N)
(Magma) [&+[2^k*3^(n-3*k)*Binomial(n-3*k+6, 6)*Binomial(2*(n-3*k), 2*k): k in [0..Floor(n/4)]] : n in [0..30] ]; // Vincenzo Librandi, Dec 21 2025

Crossrefs

Cf. A391723.

Sequence in context: A282923 A023019 A036220 * A022649 A165094 A365849

Adjacent sequences: A391721 A391722 A391723 * A391725 A391726 A391727

Keyword

nonn

Author

Seiichi Manyama, Dec 18 2025

Status

approved