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Triangle read by rows related to double factorial of odd numbers (A001147).
0

%I #4 Oct 30 2013 04:52:29

%S 1,0,1,1,0,2,0,11,0,4,25,0,72,0,8,0,589,0,340,0,16,2025,0,7010,0,1328,

%T 0,32,0,75319,0,55160,0,4592,0,64,342225,0,1335328,0,334752,0,14592,0,

%U 128

%N Triangle read by rows related to double factorial of odd numbers (A001147).

%C The row polynomials t(n,x):= sum(T(n,k)*x^k, k=0..n) satisfy the recurrence relation t(n,x) = 2x*t(n-1,x) + ((2n-3)^2)*t(n-2,x); t(0,x) = 1, t(1,x) = x.

%F T(n,k) = 2*T(n-1,k-1) + ((2n-3)^2)*T(n-2,k); T(0,0) = 1, T(1,0) = 0, T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.

%e Triangle begins:

%e 1

%e 0, 1

%e 1, 0, 2

%e 0, 11, 0, 4

%e 25, 0, 72, 0, 8

%e 0, 589, 0, 340, 0, 16

%e 2025, 0, 7010, 0, 1328, 0, 32

%e 0, 75319, 0, 55160, 0, 4592, 0, 64

%Y T(2n,0) = A007696(n)^2.

%Y T(n,n) = A011782(n).

%Y A001147 (row sums).

%Y Cf. A060524 (similar sequence).

%K easy,nonn,tabl

%O 0,6

%A _Philippe Deléham_, Oct 27 2013