OFFSET
0,2
COMMENTS
S3(n,x) := Sum_{k=0..n} S(n-k,x)*S2(k,x) = Sum_{m=0..floor(n/2)} a(n,m)*x^(n-2*m) with the second convolution S2(n,x) given by array A128503.
Row polynomials P3(n,x) := Sum_{m=0..floor(n/2)} a(n,m)*x^m (increasing powers of x).
LINKS
Wolfdieter Lang, First 15 rows and more.
FORMULA
a(n,m) = binomial(n-m+3,3)*binomial(n-m,m)*(-1)^m, m = 0..floor(n/2), n >= 0.
a(n,m) = binomial(m+3,3)*binomial(n-m+3,m+3)*(-1)^m, m = 0..floor(n/2), n >= 0.
G.f. for S3(n,x): 1/(1-x*z+z^2)^4.
G.f. for P3(n,x): 1/(1-z+x*z^2)^4.
EXAMPLE
1;
4;
10, -4;
20, -20;
35, -60, 10;
56, -140, 60;
84, -280, 210, -20;
120,-504, 560, -140;
...
n=4: [35,-60,10] stands also for the row polynomial P3(4,x) = 35-60*x+10*x^2.
CROSSREFS
KEYWORD
sign,tabf,easy
AUTHOR
Wolfdieter Lang, Apr 04 2007
EXTENSIONS
Name edited by Petros Hadjicostas, Sep 04 2019
STATUS
approved