A110518 - OEIS

%I #18 Aug 30 2017 09:37:51

%S 1,0,1,0,3,1,0,18,6,1,0,135,45,9,1,0,1134,378,81,12,1,0,10206,3402,

%T 756,126,15,1,0,96228,32076,7290,1296,180,18,1,0,938223,312741,72171,

%U 13365,2025,243,21,1,0,9382230,3127410,729729,138996,22275,2970,315,24,1,0

%N Riordan array (1, x*c(3x)), c(x) the g.f. of A000108.

%C Row sums are C(3;n), A064063. Inverse is A110517. Diagonal sums are A110525.

%H G. C. Greubel, <a href="/A110518/b110518.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)*3^(n-k), n > 0, k > 0.

%F T(n,k) = A106566(n,k)*3^(n-k). - _Philippe Deléham_, Nov 08 2007

%F Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 3, 3, 3, 3, 3, 3, 3, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Sep 23 2014

%e Rows begin

%e   1;

%e   0,    1;

%e   0,    3,    1;

%e   0,   18,    6,    1;

%e   0,  135,   45,    9,    1;

%e   0, 1134,  378,   81,   12,    1;

%e   ...

%e Production matrix begins:

%e   0,   1;

%e   0,   3,   1;

%e   0,   9,   3,   1;

%e   0,  27,   9,   3,   1;

%e   0,  81,  27,   9,   3,   1;

%e   0, 243,  81,  27,   9,   3,   1;

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

%t T[0, 0] := 1; T[0, k_] := 0; T[n_, k_] := (k/n)*3^(n - k)*Binomial[2*n - k - 1, n - k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Aug 29 2017 *)

%o (PARI) concat([1], for(n=1,10, for(k=0,n, print1((k/n)*3^(n-k)*binomial(2*n-k-1,n-k), ", ")))) \\ _G. C. Greubel_, Aug 29 2017

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Jul 24 2005