A382496 a(n) = Sum_{k=0..floor(n/3)} (k+1) * binomial(2*k,2*n-6*k).

1, 0, 0, 2, 2, 0, 3, 18, 3, 4, 60, 60, 9, 140, 350, 146, 275, 1260, 1267, 732, 3471, 6476, 4193, 8470, 24040, 25104, 24388, 72810, 117368, 102672, 202031, 440750, 490884, 612012, 1419042, 2121626, 2281049, 4267188, 7951185, 9511604, 13402924, 26600984, 38465043, 47376620

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..5000 (terms 0..1500 from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (0,0,4,4,0,-6,-4,-6,4,-4,-4,3,4,-6,4,-1).

Formula

G.f.: ((1-x^3-x^4)^2 + 4*x^7) / ((1-x^3-x^4)^2 - 4*x^7)^2.

a(n) = 4*a(n-3) + 4*a(n-4) - 6*a(n-6) - 4*a(n-7) - 6*a(n-8) + 4*a(n-9) - 4*a(n-10) - 4*a(n-11) + 3*a(n-12) + 4*a(n-13) - 6*a(n-14) + 4*a(n-15) - a(n-16).

Mathematica

Table[Sum[(k+1)*Binomial[2*k, 2*n-6*k], {k, 0, Floor[n/3]}], {n, 0, 43}] (* Vincenzo Librandi, May 12 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (k+1)*binomial(2*k, 2*n-6*k));
(PARI) my(N=1, M=50, x='x+O('x^M), X=1-x^3-x^4, Y=7); 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) [&+[(k+1)*Binomial(2*k, 2*n-6*k): k in [0..n]]: n in [0..45]]; // Vincenzo Librandi, May 12 2025

Crossrefs

Cf. A381421, A382300.

Cf. A034839, A375470, A390781.

Sequence in context: A244129 A363907 A342987 * A090657 A167001 A108563

Adjacent sequences: A382493 A382494 A382495 * A382497 A382498 A382499

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 29 2025

Status

approved