A342170 Product of first n little Schröder numbers.

1, 1, 3, 33, 1485, 292545, 264168135, 1130375449665, 23503896724884345, 2422053053602606867905, 1256704025339194996874320395, 3326147448057830199712191898815585, 45398150793225628820115544929795174823365, 3225056167710201318911738099365978237877235350145

Offset: 0

Author

Vaclav Kotesovec, Mar 03 2021

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Super Catalan Number.
• Wikipedia, Schroeder-Hipparchus numbers.

Crossrefs

Cf. A001003, A003046, A003266, A070825, A294373, A342166.

Programs

Maple

b:= proc(n) option remember; `if`(n<2, 1,
      ((6*n-3)*b(n-1)-(n-2)*b(n-2))/(n+1))
    end:
a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 03 2021

Mathematica

Table[Product[Hypergeometric2F1[1-k, k+2, 2, -1], {k, 1, n}], {n, 0, 15}]
FoldList[Times, 1, Table[Hypergeometric2F1[1 - n, n + 2, 2, -1], {n, 1, 15}]]

Formula

a(n) = Product_{k=1..n} A001003(k).

a(n) ~ c * (1 + sqrt(2))^(n*(n+2)) * exp(3*n/2) / (2^((7*n + 3)/4) * Pi^((2*n + 3)/4) * n^(3*n/2 + 3/2 + 9/(16*sqrt(2)))), where c = 0.89405100528141459535141257102427907468205556782800836208733677564241771912...