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A110510 Riordan array (1, x*c(2x)), c(x) the g.f. of A000108.

%I A110510 #24 Feb 02 2020 16:58:24
%S A110510 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 A110510 10,1,0,8448,4224,1440,384,80,12,1,0,54912,27456,9504,2640,600,108,14,
%U A110510 1,0,366080,183040,64064,18304,4400,880,140,16,1,0,2489344,1244672,439296
%N A110510 Riordan array (1, x*c(2x)), c(x) the g.f. of A000108.
%C A110510 Row sums are C(2;n), A064062. Inverse is A110509. Diagonal sums are A108308. [Corrected by _Philippe Deléham_, Nov 09 2007]
%C A110510 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 A110510 G. C. Greubel, <a href="/A110510/b110510.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A110510 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 A110510 T(n,k) = A106566(n,k)*2^(n-k). - _Philippe Deléham_, Nov 08 2007
%F A110510 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 A110510 Rows begin
%e A110510   1;
%e A110510   0,   1;
%e A110510   0,   2,   1;
%e A110510   0,   8,   4,   1;
%e A110510   0,  40,  20,   6,   1;
%e A110510   0, 224, 112,  36,   8,   1;
%e A110510   ...
%e A110510 Production matrix begins:
%e A110510   0,  1;
%e A110510   0,  2,  1;
%e A110510   0,  4,  2,  1;
%e A110510   0,  8,  4,  2,  1;
%e A110510   0, 16,  8,  4,  2,  1;
%e A110510   0, 32, 16,  8,  4,  2,  1;
%e A110510   0, 64, 32, 16,  8,  4,  2,  1;
%e A110510   ... - _Philippe Deléham_, Sep 23 2014
%t A110510 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 A110510 (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 A110510 easy,nonn,tabl
%O A110510 0,5
%A A110510 _Paul Barry_, Jul 24 2005