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Triangle read by rows: T(n,k) = Fibonacci(k+1)*binomial(n,k) + (k+1)*binomial(n,k+1) (0 <= k <= n).
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%I #13 Nov 14 2019 09:42:17

%S 1,2,1,3,4,2,4,9,9,3,5,16,24,16,5,6,25,50,50,30,8,7,36,90,120,105,54,

%T 13,8,49,147,245,280,210,98,21,9,64,224,448,630,616,420,176,34,10,81,

%U 324,756,1260,1512,1344,828,315,55,11,100,450,1200,2310,3276,3570,2880,1620

%N Triangle read by rows: T(n,k) = Fibonacci(k+1)*binomial(n,k) + (k+1)*binomial(n,k+1) (0 <= k <= n).

%C Binomial transform of the bidiagonal matrix with the Fibonacci numbers (1, 1, 2, 3, 5, 8, ...) in the main diagonal and (1, 2, 3, ...) in the subdiagonal.

%C Sum of terms in row n = n*2^(n-1) + Fibonacci(2n+1) (A081663).

%e First few rows of the triangle:

%e 1;

%e 2, 1;

%e 3, 4, 2;

%e 4, 9, 9, 3;

%e 5, 16, 24, 16, 5;

%e 6, 25, 50, 50, 30, 8;

%e 7, 36, 90, 120, 105, 54, 13;

%e 8, 49, 147, 245, 280, 210, 98, 21;

%e ...

%p with(combinat): T:=(n,k)->binomial(n,k)*fibonacci(k+1)+(k+1)*binomial(n,k+1): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

%Y Cf. A081663, A081659.

%Y Cf. A000045.

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Nov 20 2006

%E Edited by _N. J. A. Sloane_, Nov 29 2006