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A342178 Product of first n central Delannoy numbers.

%I A342178 #14 Feb 16 2025 08:34:01
%S A342178 1,3,39,2457,788697,1327377051,11931792311439,580350446236081521,
%T A342178 154215943727867706493809,225550533306461376412704772467,
%U A342178 1826384842574005591817185497927226551,82272644789290466599017454496002856892236169
%N A342178 Product of first n central Delannoy numbers.
%H A342178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DelannoyNumber.html">Delannoy Number</a>.
%F A342178 a(n) = Product_{k=1..n} A001850(k).
%F A342178 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...
%p A342178 b:= proc(n) option remember; `if`(n<1, 1,
%p A342178       (3*(2*n-1)*b(n-1) -(n-1)*b(n-2))/n)
%p A342178     end:
%p A342178 a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:
%p A342178 seq(a(n), n=0..15);  # _Alois P. Heinz_, Mar 04 2021
%t A342178 Table[Product[Hypergeometric2F1[-k, k+1, 1, -1], {k, 1, n}], {n, 0, 15}]
%t A342178 FoldList[Times, 1, Table[Hypergeometric2F1[-n, n + 1, 1, -1], {n, 1, 15}]]
%o A342178 (PARI) D(n) = sum(k=0, n, binomial(n, k)*binomial(n+k, k)); \\ A001850
%o A342178 a(n) = prod(k=0, n, D(k)); \\ _Michel Marcus_, Mar 04 2021
%Y A342178 Cf. A001850, A342170, A342177.
%K A342178 nonn
%O A342178 0,2
%A A342178 _Vaclav Kotesovec_, Mar 04 2021