OFFSET
0,2
COMMENTS
S1(n,x):=sum(S(n-k,x)*S(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)).
The unsigned column sequences, m>=0, divided by (m+1) give Pascal triangle column sequences for m+1.
G.f. for column m sequence: ((-1)^m)*(m+1)*(x^(2*m))/(1-x)^(m+2), m>=0.
Row polynomials P1(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).
Written as a triangle with increasing powers of x this is A294519. - Wolfdieter Lang, Nov 12 2017
LINKS
FORMULA
a(n,m)=binomial(n-m,m)*(n+1-m)*(-1)^m, m=0..floor(n/2), n>=0.
a(n,m)=binomial(n+1-m,m+1)*(m+1)*(-1)^m, m=0..floor(n/2), n>=0.
G.f. for S1(n,x): 1/(1-x*z+z^2)^2.
G.f. for P1(n,x): 1/(1-z+x*z^2)^2.
EXAMPLE
[1];[2];[3,-2],[4,-6];[5,-12,3];[6,-20,12];[7,-30,30,-4];[8,-42,60,-20];...
n=4: [5,-12,3] stands for the polynomial S1(4,x) = 5*x^4-12*x^2+3 = 2*(S(4,x)*1+S(3,x)*S(1,x))+S(2,x)*S(2,x).
n=4: [5,-12,3] stands also for the row polynomial P1(4,x) = 5-12*x+3*x^2.
CROSSREFS
KEYWORD
sign,tabf,easy
AUTHOR
Wolfdieter Lang Apr 04 2007
STATUS
approved