A262664 Expansion of (1-2*x)/((2-x)*sqrt(5*x^2-6*x+1))+1/(2-x).

1, 1, 3, 13, 59, 271, 1257, 5881, 27715, 131395, 626033, 2995147, 14380181, 69249337, 334345091, 1617924973, 7844900339, 38105139907, 185380469961, 903147125143, 4405621159969, 21515837558557, 105188202097091, 514747668977263

Offset: 0

Author

Vladimir Kruchinin, Sep 26 2015

Keywords

• nonn

Links

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

Crossrefs

Cf. A033321.

Programs

Mathematica

Table[Sum[Binomial[n, k] Sum[2^i Binomial[k, n - k - i] Binomial[k + i - 1, i] (-1)^(n - k - i), {i, 0, n - k}], {k, 0, n}], {n, 0, 23}] (* Michael De Vlieger, Sep 26 2015 *)

Maxima

a(n):=sum(binomial(n,k)*sum(2^i*binomial(k,n-k-i)*binomial(k+i-1,i)*(-1)^(n-k-i),i,0,n-k),k,0,n);

PARI

x='x+O('x^50); Vec((1-2*x)/((2-x)*sqrt(5*x^2-6*x+1))+1/(2-x)) \\ G. C. Greubel, Jun 04 2017

Formula

a(n) = Sum_{k=0..n}(binomial(n,k)*Sum_{i=0..n-k}(2^i*binomial(k,n-k-i)*binomial(k+i-1,i)*(-1)^(n-k-i))).

G.f.: A(x) = x*B'(x)/B(x), where B(x)/x is g.f. of A033321.

a(n) ~ 5^(n+1/2)/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 29 2015

D-finite with recurrence: 2*n*(3*n-4)*a(n) = (39*n^2 - 70*n + 28)*a(n-1) - (48*n^2 - 103*n + 34)*a(n-2) + 5*(n-2)*(3*n-1)*a(n-3). - Vaclav Kotesovec, Sep 29 2015