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A067979
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Triangle of incomplete convolutions of Lucas numbers L(n+1) := A000204(n+1), n>=0.
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10
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1, 3, 6, 4, 13, 17, 7, 19, 31, 38, 11, 32, 48, 69, 80, 18, 51, 79, 107, 140, 158, 29, 83, 127, 176, 220, 274, 303, 47, 134, 206, 283, 360, 432, 519, 566, 76, 217, 333, 459, 580, 706, 822, 963, 1039, 123, 351, 539, 742
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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(L(k+1)*L(d+n+1-k),k=0..n).
The diagonals give A004799(n-1), A067980-7 for d=n-m= 0..8, respectively. Row sums give A067989.
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).
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FORMULA
| a(n, m)= sum(L(k+1)*L(n-k+1), k=0..m), n>=m>=0, else 0.
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).
G.f. for diagonals d= n-m>=0: (x^d)*(L(d+1)+L(d)*x)*(1-2*x)/(1-x-x^2)^2.
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EXAMPLE
| {1}; {3,6}; {4,13,17}; {7,19,31,38}; ... p(2,x)=4+13*x+17*x^2.
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CROSSREFS
| Cf. A067990 (triangle with rows read backwards).
Sequence in context: A169854 A098383 A162523 * A091808 A128719 A145691
Adjacent sequences: A067976 A067977 A067978 * A067980 A067981 A067982
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 15 2002
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