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 A078718 #44 Apr 21 2025 03:29:41
%S A078718 0,1,3,-5,14,-45,154,-546,1980,-7293,27170,-102102,386308,-1469650,
%T A078718 5616324,-21544100,82907640,-319929885,1237518450,-4796857230,
%U A078718 18627909300,-72457790790,282257178060,-1100982015900,4299680491080,-16809921068850,65785111513524,-257683159276956
%N A078718 a(n) = (-1)^n*(2*n - 1)*CatalanNumber(n - 2) for n >= 2, a(n) = n for n = 0, 1.
%C A078718 Original definition: Let f(i, j) = Sum_{k=0..2*i} binomial(2*i, k)*binomial(2*j, i+j-k)*(-1)^(i+j-k) (this is essentially the same as the triangle in A068555); then a(n) = f(n, n-2)/2. Apart perhaps from signs, f(n, 0) and f(n, n) give A000984, f(n, 1) gives A002420, f(n, n-1) gives 2*A000984.
%H A078718 G. C. Greubel, <a href="/A078718/b078718.txt">Table of n, a(n) for n = 0..1000</a>
%H A078718 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A078718 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%F A078718 G.f.: x^2*A'(x)*G(A(x))/A(x), where A(x) = x*(1+sqrt(1+4*x))/2, G(x) =(2*cosh(asinh((3^(3/2)*sqrt(x))/2)/3)*sinh(asinh((3^(3/2)*sqrt(x))/2)/3)*sqrt(x))/(sqrt(3)*sqrt((27*x)/4+1))+(4*sinh(asinh((3^(3/2)*sqrt(x))/2)/3)^2)/3+1. - _Vladimir Kruchinin_, Dec 16 2016
%F A078718 G.f.: (sqrt(4*x+1)*(4*x^2+x)+6*x^2+x)/(sqrt(4*x+1)+4*x+1). - _Vladimir Kruchinin_, Dec 17 2016
%F A078718 a(n) ~ (-1)^n * 2^(2*n-3) / sqrt(Pi*n). - _Vaclav Kotesovec_, Dec 17 2016
%F A078718 a(n) = (1/2)*Sum_{k=0,..,2*n} ( binomial(2*n, k)*binomial(2*(n - 2), 2*n - 2 - k)*(-1)^(2*n - 2 - k) ), with a(0)=0, a(1)=1. - _G. C. Greubel_, Feb 16 2017
%F A078718 D-finite with recurrence (-n+1)*a(n) +6*(-2*n+5)*a(n-1) +16*(-2*n+7)*a(n-2)=0. - _R. J. Mathar_, Nov 22 2024
%t A078718 s = (Sqrt[4*x+1]*(4*x^2+x) + 6*x^2 + x)/(Sqrt[4*x+1] + 4*x+1) + O[x]^28; CoefficientList[s, x] (* _Jean-François Alcover_, Dec 17 2016, after _Vladimir Kruchinin_ *)
%t A078718 Table[(1/2)*Sum[ Binomial[2*n, k]*Binomial[2*(n - 2), 2*n - 2 - k]*(-1)^(2*n - 2 - k), {k, 0, 2*n}], {n, 0, 50}] (* _G. C. Greubel_, Feb 16 2017 *)
%t A078718 (* Mathematica returns CatalanNumber[-2] = 0 and CatalanNumber[-1] = -1. This extended definition is in accordance with the alternative definition of the Catalan numbers C(n) = binomial(2*n, n) - binomial(2*n, n-1). *)
%t A078718 a[n_] := (-1)^n (2 n - 1) CatalanNumber[n - 2];
%t A078718 Table[a[n], {n, 0, 21}] (* _Peter Luschny_, Nov 28 2021 *)
%o A078718 (Maxima)
%o A078718 A(x) := x*(1 + sqrt(1+4*x))/2;
%o A078718 G(x) := (2*cosh(asinh((3^(3/2)*sqrt(x))/2)/3)*sinh(asinh((3^(3/2)*sqrt(x))/2)/3)* sqrt(x))/(sqrt(3)*sqrt((27*x)/4+1))+(4*sinh(asinh((3^(3/2)*sqrt(x))/2)/3)^2)/3 + 1;
%o A078718 taylor(x^2*diff(A(x),x)*G(A(x))/A(x),x,0,20); /* _Vladimir Kruchinin_, Dec 16 2016 */
%o A078718 (PARI) concat([0,1], for(n=2,25, print1(sum(k=0,2*n, (1/2)* binomial(2*n,k)* binomial(2*( n-2),2*n-k-2)*(-1)^(2*n-k-2)), ", "))) \\ _G. C. Greubel_, Feb 16 2017
%Y A078718 Cf. A000108, A068555, A000984, A002420.
%K A078718 sign
%O A078718 0,3
%A A078718 _N. J. A. Sloane_, Dec 20 2002
%E A078718 New name by _Peter Luschny_, Nov 28 2021