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A067990 Triangle A067979 with rows read backwards. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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 columns give A004799(n-1), A067980-7 for 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(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)}).
LINKS
FORMULA
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
EXAMPLE
{1}; {6,3}; {17,13,4}; {38,31,19,7}; ...; p(2,x)=17+13*x+4*x^2.
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A328011 A231881 A229005 * A174012 A050008 A166450
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Feb 15 2002
STATUS
approved

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Last modified June 19 18:58 EDT 2024. Contains 373507 sequences. (Running on oeis4.)