A342177 Product of first n Motzkin numbers.

1, 1, 2, 8, 72, 1512, 77112, 9793224, 3163211352, 2641281478920, 5779123875876960, 33507360232334614080, 519732664563742198994880, 21743016022024154894950804800, 2470745882646692817332839752643200, 767344490265348681664694707657903910400

Offset: 0

Author

Vaclav Kotesovec, Mar 04 2021

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Motzkin Number.

Crossrefs

Cf. A001006, A342170.

Programs

Maple

b:= proc(n) option remember; `if`(n<2, 1,
      ((2*n+1)*b(n-1) +(3*n-3)*b(n-2))/(n+2))
    end:
a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 04 2021

Mathematica

Table[Product[Hypergeometric2F1[(1-k)/2, -k/2, 2, 4], {k, 1, n}], {n, 0, 15}]
FoldList[Times, 1, Table[Hypergeometric2F1[(1 - n)/2, -n/2, 2, 4], {n, 1, 15}]]

Formula

a(n) = Product_{k=1..n} A001006(k).

a(n) ~ c * 3^(n*(n+4)/2) * exp(3*n/2) / (2^(n + 3/4) * Pi^(n/2 + 3/4) * n^(3*n/2 + 51/16)), where c = 1.88710807429950713889649869555827071385924787813206239984338608598135477864...