login
Triangle of coefficients of polynomials (rising powers) useful for convolutions of A001333(n+1), n >= 0 (associated Pell numbers).
1

%I #4 Mar 31 2012 13:20:05

%S 0,1,2,20,36,16,456,944,672,160,14304,33760,28800,10880,1536,575040,

%T 1466752,1413120,666880,157440,14848,27659520,74774784,79278080,

%U 43330560,13153280,2128896,143360,1548126720

%N Triangle of coefficients of polynomials (rising powers) useful for convolutions of A001333(n+1), n >= 0 (associated Pell numbers).

%C The row polynomials pPL1(n,x) := sum(a(n,m)*x^m,m=0..n) and pPL2(n,x) := sum(A062134(n,m)*x^m,m=0..n) appear in the k-fold convolution of the associated Pell numbers PL(n) := A001333(n+1), n >= 0, as follows: PL(k; n) := A054458(n+k,k)= (2*pPL1(k,n)*PL(n+1)+pPL2(k,n)*PL(n)/(k!*8^k), k >= 0.

%e {0}; {1,2}; {20,36,16}; {456,944,672,160}; ...

%e pPL1(2,n)=4*(5+9*n+4*n^2)=4*(1+n)*(5+4*n); pL2(2,n)=8*(1+3*n+2*n^2)=8*(1+n)*(1+2*n); PL(2; n)= A054460(n)=(1+n)*((5+4*n)*PL(n+1)+(1+2*n)*PL(n))/16.

%Y A062134(n, m) (companion triangle), A054458(n, m) (convolution triangle).

%K nonn,tabl

%O 0,3

%A _Wolfdieter Lang_, Jun 19 2001