OFFSET
1,1
COMMENTS
S(3,n) := Sum_{j=0..n} j^3*binomial(2*j,j)*binomial(2*(n-j),n-j).
a(n) = 2^3*S(3,n)/4^n, n >= 1.
O.g.f. for S(3,n) is G(k=3,x). See triangle A142963 for the general G(k,x) formula.
The author was led to compute such sums by a question asked by M. Greiter, Jun 27 2008.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = n^2*(3+5*n)/2.
a(n) = (2^3)*S(3,n)/4^n with the convolution S(3,n) defined above.
G.f.: x*(4+10*x+x^2)/(1-x)^4. - Joerg Arndt, Jul 02 2023
MATHEMATICA
Rest@ CoefficientList[Series[x (4 + 10 x + x^2)/(1 - x)^4, {x, 0, 39}], x] (* Michael De Vlieger, Jul 02 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 15 2008
STATUS
approved