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A061189 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A000204(n+1), n >= 0 (Lucas numbers).

%I A061189 #11 Apr 20 2025 03:23:54
%S A061189 1,2,0,-10,15,25,30,475,450,125,6000,8500,6250,5000,1250,96000,146250,
%T A061189 189375,159375,65625,9375,180000,5355000,8881250,5578125,2515625,
%U A061189 721875,78125,44100000,254700000,341775000
%N A061189 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A000204(n+1), n >= 0 (Lucas numbers).
%C A061189 The row polynomials pL2(n,x) := Sum_{m=0..n} a(n,m)*x^m and pL1(n,x) := Sum_{m=0..n} A061188(n,m)*x^m appear in the k-fold convolution of the Lucas numbers L(n+1) = A000204(n+1) = A000032(n+1), n >= 0, as follows: L(k; n) := A060922(n+k,k) = (pL1(k,n)*L(n+2)+pL2(k,n)*L(n+1))/(k!*5^k).
%e A061189 Triangle begins:
%e A061189   {1};
%e A061189   {2,0};
%e A061189   {-10,15,25};
%e A061189   {30,475,450,125};
%e A061189   ...;
%e A061189 pL2(2,n) = 5*(-2+3*n+5*n^2) = 5*(1+n)*(-2+5*n).
%e A061189 L(2; n) := A060922(n+2,2) = A060929(n) = (1+n)*((4+5*n)*L(n+2)+(-2+5*n)*L(n+1))/(2*5).
%Y A061189 Cf. A061188(n, m) (companion triangle), A060922(n, m) (Lucas convolution triangle).
%K A061189 sign,tabl,more
%O A061189 0,2
%A A061189 _Wolfdieter Lang_, Apr 20 2001