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Riordan array (1, x*c(2x)), c(x) the g.f. of A000108.
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%I #24 Feb 02 2020 16:58:24

%S 1,0,1,0,2,1,0,8,4,1,0,40,20,6,1,0,224,112,36,8,1,0,1344,672,224,56,

%T 10,1,0,8448,4224,1440,384,80,12,1,0,54912,27456,9504,2640,600,108,14,

%U 1,0,366080,183040,64064,18304,4400,880,140,16,1,0,2489344,1244672,439296

%N Riordan array (1, x*c(2x)), c(x) the g.f. of A000108.

%C Row sums are C(2;n), A064062. Inverse is A110509. Diagonal sums are A108308. [Corrected by _Philippe Deléham_, Nov 09 2007]

%C Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 2, 2, 2, 2, 2, 2, 2, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Sep 23 2014

%H G. C. Greubel, <a href="/A110510/b110510.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F Number triangle: T(0,k) = 0^k, T(n,k) = (k/n)*C(2n-k-1, n-k)*2^(n-k), n, k > 0.

%F T(n,k) = A106566(n,k)*2^(n-k). - _Philippe Deléham_, Nov 08 2007

%F T(n,k) = 2*T(n,k+1) + T(n-1,k-1) with T(n,n) = 1 and T(n,0) = 0 for n >= 1. - _Peter Bala_, Feb 02 2020

%e Rows begin

%e 1;

%e 0, 1;

%e 0, 2, 1;

%e 0, 8, 4, 1;

%e 0, 40, 20, 6, 1;

%e 0, 224, 112, 36, 8, 1;

%e ...

%e Production matrix begins:

%e 0, 1;

%e 0, 2, 1;

%e 0, 4, 2, 1;

%e 0, 8, 4, 2, 1;

%e 0, 16, 8, 4, 2, 1;

%e 0, 32, 16, 8, 4, 2, 1;

%e 0, 64, 32, 16, 8, 4, 2, 1;

%e ... - _Philippe Deléham_, Sep 23 2014

%t T[n_, k_] := (k/n)*Binomial[2*n - k - 1, n - k]*2^(n - k); Join[{1}, Table[T[n, k], {n, 1, 10}, {k, 0, n}]] // Flatten (* _G. C. Greubel_, Aug 29 2017 *)

%o (PARI) concat([1], for(n=1,25, for(k=0,n, print1((k/n)*binomial(2*n-k-1, n-k)*2^(n-k), ", ")))) \\ _G. C. Greubel_, Aug 29 2017

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Jul 24 2005