A342178 Product of first n central Delannoy numbers.

1, 3, 39, 2457, 788697, 1327377051, 11931792311439, 580350446236081521, 154215943727867706493809, 225550533306461376412704772467, 1826384842574005591817185497927226551, 82272644789290466599017454496002856892236169

Offset: 0

Author

Vaclav Kotesovec, Mar 04 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A001850, A342170, A342177.

Programs

Maple

b:= proc(n) option remember; `if`(n<1, 1,
      (3*(2*n-1)*b(n-1) -(n-1)*b(n-2))/n)
    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 04 2021

Mathematica

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

PARI

D(n) = sum(k=0, n, binomial(n, k)*binomial(n+k, k)); \\ A001850
a(n) = prod(k=0, n, D(k)); \\ Michel Marcus, Mar 04 2021

Formula

a(n) = Product_{k=1..n} A001850(k).

a(n) ~ c * (1 + sqrt(2))^(n*(n+2)) * exp(n/2) / (2^((5*n+1)/4) * Pi^(n/2 + 1/4) * n^((n+1)/2 - 3/(16*sqrt(2)))), where c = 0.9486848745280397752870611535632702994491680306036912732565033220352175749...