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A067979 Triangle read by rows of incomplete convolutions of Lucas numbers L(n+1) = A000204(n+1), n>=0.

%I A067979 #20 Dec 18 2017 02:35:27
%S A067979 1,3,6,4,13,17,7,19,31,38,11,32,48,69,80,18,51,79,107,140,158,29,83,
%T A067979 127,176,220,274,303,47,134,206,283,360,432,519,566,76,217,333,459,
%U A067979 580,706,822,963,1039,123,351,539,742,940,1138,1341,1529,1757,1880,199,568,872,1201,1520,1844,2163,2492,2796,3165,3364
%N A067979 Triangle read by rows of incomplete convolutions of Lucas numbers L(n+1) = A000204(n+1), n>=0.
%C A067979 The diagonals d>=0 (d=0: main diagonal) give convolutions of Lucas numbers L(n+1) := A000204(n+1), n>=0, with those with d-shifted index: a(d+n,d) = Sum_{k=0..n} L(k+1)*L(d+n+1-k).
%C A067979 The diagonals give A004799(n-1), A067980-7 for d=n-m= 0..8, respectively. Row sums give A067989.
%C A067979 The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(x*z)*(A(z)-x*A(x*z))/(1-x), with A(x) := (1+2*x)/(1-x-x^2) (g.f. Lucas L(n+1), n>=0).
%H A067979 Michael De Vlieger, <a href="/A067979/b067979.txt">Table of n, a(n) for n = 0..10000</a>
%F A067979 a(n, m) = Sum_{k=0..m} L(k+1)*L(n-k+1), n>=m>=0, else 0.
%F A067979 a(n, m) = (m+1)*L(n-m+1)*F(m) + ((m+1)*L(n-m+1) + m*L(n-m))*F(m+1), n>=m>=0, with F(n) := A000045(n) (Fibonacci) and L(n) := A000032(n) (Lucas).
%F A067979 G.f. for diagonals d= n-m>=0: (x^d)*(L(d+1)+L(d)*x)*(1-2*x)/(1-x-x^2)^2.
%F A067979 a(n, m) = -(-1)^m*F(n-2*m-1) + m*L(n+2)+F(n+3), with F(-n) = (-1)^(n+1) * F(n), hence a(n, m) = -5*A067330(n, m)+2*(m+1)*L(n+2), n>=m>=0. - _Ehren Metcalfe_, Apr 11 2016
%e A067979 Triangle begins:
%e A067979   {1};
%e A067979   {3,6};
%e A067979   {4,13,17};     p(2,x) = 4+13*x+17*x^2
%e A067979   {7,19,31,38};
%e A067979   ...
%t A067979 Table[Sum[LucasL[k + 1] LucasL[n - k + 1], {k, 0, m}], {n, 0, 10}, {m, 0, n}] // Flatten (* _Michael De Vlieger_, Apr 11 2016 *)
%o A067979 (PARI) for(n=0,10, for(k=0,n, print1(sum(m=0,k,(fibonacci(m+2) + fibonacci(m))*(fibonacci(n-m+2) + fibonacci(n-m))), ", "))) \\ _G. C. Greubel_, Dec 17 2017
%Y A067979 Cf. A067990 (triangle with rows read backwards).
%K A067979 nonn,easy,tabl
%O A067979 0,2
%A A067979 _Wolfdieter Lang_, Feb 15 2002