A378804 a(n) = n * 2^n * binomial(4*n, n).

0, 8, 224, 5280, 116480, 2480640, 51684864, 1060899840, 21541478400, 433812234240, 8680043806720, 172774871965696, 3424347806171136, 67626404043161600, 1331466198928588800, 26145958720005734400, 512257621575157678080, 10016204637370583089152, 195501127311163895316480

Offset

0,2

Links

Amiram Eldar, Table of n, a(n) for n = 0..500

Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36.

Formula

a(n) = A036289(n) * A005810(n).

a(n) = 2^n * A378802(n).

a(n) == 0 (mod 8).

Sum_{n>=1} (-1)^n/a(n) = (log(2) - 6*log(3))/7 + Sum_{r: 2*r^3 + 12*r + 13 = 0} log(r+2)/(r+3) = -0.120716907732393305... (Borwein and Girgensohn, 2005, p. 32, eq. (43)).

a(n) ~ 2^(9*n+1/2) * sqrt(n) / (3^(3*n+1/2) * sqrt(Pi)). - Amiram Eldar, Sep 23 2025

Mathematica

a[n_] := n * 2^n * Binomial[4*n, n]; Array[a, 20, 0]

Prog

(PARI) a(n) = n * 2^n * binomial(4*n, n);

Crossrefs

Cf. A005810, A036289, A378802.

Sequence in context: A301360 A267235 A230144 * A013377 A214383 A220546

Adjacent sequences: A378801 A378802 A378803 * A378805 A378806 A378807

Keyword

nonn,easy

Author

Amiram Eldar, Dec 07 2024

Status

approved