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).

0, 1, 3, 11, 45, 191, 833, 3695, 16593, 75199, 343233, 1575551, 7265921, 33637631, 156234497, 727681791, 3397475585, 15896054783, 74512968705, 349859309567, 1645121398785, 7746058698751, 36516283891713, 172332643868671, 814108326764545, 3849410342715391

Offset: 0

Author

Vladimir Kruchinin, Mar 01 2014

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A052709.

Programs

Mathematica

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 *)
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 *)

Maxima

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);

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

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

Formula

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).

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.

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

a(n) ~ (2 + 2*sqrt(2))^n / (2^(5/4) * sqrt(1+sqrt(2)) * sqrt(Pi*n)). - Vaclav Kotesovec, Mar 03 2014