This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A073617 #39 Aug 04 2025 05:40:36
%S A073617 1,1,1,2,3,12,30,240,1050,16800,132300,4233600,61122600,3911846400,
%T A073617 104886381600,13425456844800,674943865596000,172785629592576000,
%U A073617 16407885372638760000,8400837310791045120000,1515727634953623371280000,1552105098192510332190720000
%N A073617 Consider Pascal's triangle A007318; a(n) = product of terms at +45 degrees slope with the horizontal.
%C A073617 The sum of the terms pertaining to the above product is the (n+1)-th Fibonacci number: 1 + 5 + 6 + 1 = 13.
%C A073617 n divides A073617(n+1) for n>=1; see the Mathematica section. [_Clark Kimberling_, Feb 29 2012]
%H A073617 Vaclav Kotesovec, <a href="/A073617/b073617.txt">Table of n, a(n) for n = 0..118</a>
%F A073617 a(n) = Product_{k=0..floor(n/2)} binomial(n-k,k).
%F A073617 a(2n+1)/a(2n-1) = binomial(2n,n); a(2n)/a(2n-2) = (1/2)*binomial(2n,n); (a(2n+1)*a(2n-2))/(a(2n)*a(2n-1)) = 2. - _John Molokach_, Sep 09 2013
%F A073617 a(n) ~ A^(3/2) * 2^(n*(n+1)/4 - 1/6 + (-1)^n/4) * exp(n/4 - 1/8) / (n^((n+1)/4 + (-1)^n/8) * Pi^(n/4 + 3/8 + (-1)^n/8)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Jun 10 2025
%e A073617 For n=6 the diagonal is 1,5,6,1 and product of the terms = 30 hence a(6) = 30.
%p A073617 a:= n-> mul(binomial(n-i, i), i=0..floor(n/2)):
%p A073617 seq(a(n), n=0..21); # _Alois P. Heinz_, Nov 27 2023
%t A073617 p[n_] := Product[Binomial[n + 1 - k, k], {k, 1, Floor[(n + 1)/2]}]
%t A073617 Table[p[n], {n, 1, 20}] (* A073617(n+1) *)
%t A073617 Table[p[n]/n, {n, 1, 20}] (* A208649 *)
%t A073617 (* _Clark Kimberling_, Feb 29 2012 *)
%t A073617 (* Second program *)
%t A073617 Join[{1}, Table[If[EvenQ[n], 2^(3/2 - n/4) * Sqrt[BarnesG[n]] * Gamma[n] / (n*BarnesG[n/2]^2 * Gamma[n/2]^(7/2)), Glaisher^3 * 2^((-10 + 3*n + 6*n^2)/12) * BarnesG[n/2]^2 * Gamma[n/2]^(5/2) / (E^(1/4) * Pi^(1/4 + n/2) * Sqrt[BarnesG[n]] * Gamma[n])], {n, 1, 25}]] (* _Vaclav Kotesovec_, Jun 10 2025 *)
%Y A073617 Cf. A000045, A073618, A007685, A208649, A000984.
%K A073617 nonn
%O A073617 0,4
%A A073617 _Amarnath Murthy_, Aug 07 2002
%E A073617 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 22 2003