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A062133 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A001333(n+1), n >= 0 (associated Pell numbers).

%I A062133 #8 Apr 20 2025 03:23:31
%S A062133 0,1,2,20,36,16,456,944,672,160,14304,33760,28800,10880,1536,575040,
%T A062133 1466752,1413120,666880,157440,14848,27659520,74774784,79278080,
%U A062133 43330560,13153280,2128896,143360,1548126720
%N A062133 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A001333(n+1), n >= 0 (associated Pell numbers).
%C A062133 The row polynomials pPL1(n,x) := Sum_{m=0..n} a(n,m)*x^m, and pPL2(n,x) := Sum_{m=0..n} A062134(n,m)*x^m 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 A062133 Triangle begins:
%e A062133   {0};
%e A062133   {1,2};
%e A062133   {20,36,16};
%e A062133   {456,944,672,160};
%e A062133   ...
%e A062133 pPL1(2,n) = 4*(5+9*n+4*n^2) = 4*(1+n)*(5+4*n).
%e A062133 pPL2(2,n) = 8*(1+3*n+2*n^2) = 8*(1+n)*(1+2*n).
%e A062133 PL(2; n) = A054460(n) = (1+n)*((5+4*n)*PL(n+1)+(1+2*n)*PL(n))/16.
%Y A062133 Cf. A062134(n, m) (companion triangle), A054458(n, m) (convolution triangle).
%K A062133 nonn,tabl,more
%O A062133 0,3
%A A062133 _Wolfdieter Lang_, Jun 19 2001