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Bisection of Lucas triangle A060922: even-indexed members of column sequences of A060922 (not counting leading zeros).
9

%I #14 Mar 30 2024 13:24:17

%S 1,4,1,11,17,1,29,80,39,1,76,303,315,70,1,199,1039,1687,905,110,1,521,

%T 3364,7470,6666,2120,159,1,1364,10493,29634,37580,20965,4311,217,1,

%U 3571,31885,109421,181074,148545

%N Bisection of Lucas triangle A060922: even-indexed members of column sequences of A060922 (not counting leading zeros).

%F a(n, m) = A060922(2*n-m, m).

%F a(n, m) = ((2*(n-m)+1)*A060924(n-1, m-1) + 2*(4*n-3*m)*a(n-1, m-1) + 4*(2*n-m-1)*A060924(n-2, m-1))/(5*m), m >= n >= 1; a(n, 0)= A002878(n); else 0.

%F G.f. for column m >= 0: x^m*pLe(m+1, x)/(1-3*x+x^2)^(m+1), where pLe(n, x) := Sum_{m=0..n+floor(n/2)} A061186(n, m)*x^m are the row polynomials of the (signed) staircase A061186.

%F T(n,k) = 3*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2) + 4*T(n-3,k-2), T(0,0) = 1, T(1,0) = 4, T(1,1) = 1, T(2,0) = 11, T(2,1) = 17, T(2,2) = 1, T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Jan 21 2014

%e Triangle begins:

%e {1};

%e {4,1};

%e {11,17,1};

%e {29,80,39,1};

%e ...

%e pLe(2,x) = 1+11*x-11*x^2+4*x^3.

%Y Row sums give A060926.

%Y Column sequences (without leading zeros) are, for m=0..3: A002878, A060934-A060936.

%Y Companion triangle A060924 (odd part).

%Y Cf. A060922.

%K nonn,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Apr 20 2001