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 A238578 #32 Feb 25 2025 20:38:20
%S A238578 0,1,3,11,45,191,833,3695,16593,75199,343233,1575551,7265921,33637631,
%T A238578 156234497,727681791,3397475585,15896054783,74512968705,349859309567,
%U A238578 1645121398785,7746058698751,36516283891713,172332643868671,814108326764545,3849410342715391
%N A238578 Expansion of -(-4*x^4 + sqrt(-4*x^2-4*x+1) * (2*x^3+x^2-2*x) -12*x^3-7*x^2+2*x) / (sqrt(-4*x^2-4*x+1) * (4*x^3+8*x^2+3*x-1) - 4*x^3-8*x^2-3*x+1).
%H A238578 G. C. Greubel, <a href="/A238578/b238578.txt">Table of n, a(n) for n = 0..1000</a>
%F A238578 a(n) = Sum_{k=1..n} Sum_{i=0..(n-k)} C(k,n-k-i)*C(k+i-1,k-1)*C(n-1,k-1).
%F A238578 G.f.: A(x) = x*(F(x)-x)*F'(x)/F(x)^2, where F(x) = (1-sqrt(-4*x^2-4*x+1))/(2*x+2), F(x) is the g.f. of A052709.
%F A238578 D-finite with recurrence: (for n>5): (n-5)*(n-1)*a(n) = (3*n^2 - 20*n + 23)*a(n-1) + 2*(n-2)*(4*n-19)*a(n-2) + 4*(n-4)*(n-3)*a(n-3). - _Vaclav Kotesovec_, Mar 03 2014
%F A238578 a(n) ~ (2 + 2*sqrt(2))^n / (2^(5/4) * sqrt(1+sqrt(2)) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 03 2014
%t A238578 Table[Sum[Binomial[n - 1, k - 1] * Sum[Binomial[k, n - k - i] * Binomial[k + i - 1, k - 1], {i, 0, n - k}], {k, n}], {n, 0, 20}] (* _Wesley Ivan Hurt_, Mar 02 2014 *)
%t A238578 CoefficientList[Series[-(-4*x^4 + Sqrt[-4*x^2-4*x+1]*(2*x^3+x^2-2*x) -12*x^3-7*x^2+2*x)/(Sqrt[-4*x^2-4*x+1]*(4*x^3+8*x^2+3*x-1) - 4*x^3-8*x^2-3*x+1), {x, 0, 50}], x] (* _G. C. Greubel_, Jun 01 2017 *)
%o A238578 (Maxima)
%o A238578 a(n):= sum((sum(binomial(k,n-k-i)*binomial(k+i-1,k-1), i,0,n-k)) *binomial(n-1,k-1), k,1,n);
%o A238578 (PARI) my(x='x+O('x^50)); concat([0], Vec(-(-4*x^4 + sqrt(-4*x^2-4*x+1)*(2*x^3+x^2-2*x) -12*x^3-7*x^2+2*x)/(sqrt(-4*x^2-4*x+1)*(4*x^3+8*x^2+3*x-1) - 4*x^3-8*x^2-3*x+1))) \\ _G. C. Greubel_, Jun 01 2017
%o A238578 (PARI) for(n=0,25, print1(sum(k=1,n, binomial(n-1,k-1)*sum(i=0,n-k, binomial(k,n-k-i)*binomial(k+i-1,k-1))), ", ")) \\ _G. C. Greubel_, Jun 01 2017
%Y A238578 Cf. A052709.
%K A238578 nonn
%O A238578 0,3
%A A238578 _Vladimir Kruchinin_, Mar 01 2014