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A061188 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A000032(n+1), n >= 0 (Lucas numbers).

%I A061188 #12 Apr 20 2025 03:23:44
%S A061188 0,1,5,20,45,25,240,350,600,250,3000,9250,13125,8750,1875,93000,
%T A061188 373750,361875,240625,103125,15625,3690000,11077500,12818750,8531250,
%U A061188 4156250,1181250,125000,116550000,312037500
%N A061188 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A000032(n+1), n >= 0 (Lucas numbers).
%C A061188 The row polynomials pL1(n,x) := Sum_{m=0..n} a(n,m)*x^m and pL2(n,x) := Sum_{m=0..n} A061189(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 A061188 Triangle begins:
%e A061188   {0};
%e A061188   {1,5};
%e A061188   {20,45,25};
%e A061188   {240,350,600,250};
%e A061188   ...;
%e A061188 pL1(2,n) = 5*(4+9*n+5*n^2) = 5*(1+n)*(4+5*n).
%Y A061188 Cf. A061189(n, m) (companion triangle), A060922(n, m) (Lucas convolution triangle).
%K A061188 nonn,tabl,more
%O A061188 0,3
%A A061188 _Wolfdieter Lang_, Apr 20 2001