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A067990
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Triangle A067979 with rows read backwards.
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12
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1, 6, 3, 17, 13, 4, 38, 31, 19, 7, 80, 69, 48, 32, 11, 158, 140, 107, 79, 51, 18, 303, 274, 220, 176, 127, 83, 29, 566, 519, 432, 360, 283, 206, 134, 47, 1039, 963, 822, 706, 580, 459, 333, 217, 76, 1880, 1757, 1529, 1341, 1138, 940, 742, 539, 351, 123, 3364, 3165, 2796, 2492, 2163, 1844, 1520, 1201
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OFFSET
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0,2
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COMMENTS
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The column m (without leading 0's) gives the convolution of Lucas numbers {L(n+1) := A000204(n+1)}, n>=0, with those with m-shifted index: a(n+m,m)=sum(L(k+1)*L(m+n+1-k),k=0..n), n>=0,m=0,1,...
The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(z)*(A(z)-x*A(x*z))/(1-x), with A(x) := (1+2*x)/(1-x-x^2) (g.f. for Lucas {L(n+1)}).
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LINKS
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FORMULA
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a(n, m)=A067330(n, n-m), n>=m>=0, else 0.
a(n, m)=(n-m+1)*L(m+1)*F(n-m)+((n-m+1)*L(m+1)+(n-m)*L(m))*F(n-m+1), n>=m>=0, else 0; with F(n) := A000045(n)(Fibonacci) and L(n) := A000032(n) (Lucas).
G.f. for column m=0, 1, ...: (x^m)*(L(m+1)+L(m)*x)*(1+2*x)/(1-x-x^2)^2.
a(n, m) = -(-1)^m*F(n-2*m+1)-m*L(n+2)+n*L(n+2)+F(n+3), with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = -5*A067418(n, m)+2*(n-m+1)*L(n+2), n>=m>=0. - Ehren Metcalfe, Apr 11 2016
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EXAMPLE
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{1}; {6,3}; {17,13,4}; {38,31,19,7}; ...; p(2,x)=17+13*x+4*x^2.
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MATHEMATICA
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Reverse /@ Table[Sum[LucasL[k + 1] LucasL[n - k + 1], {k, 0, m}], {n, 0, 11}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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