A110519 Riordan array (1/(1-x*c(3*x)), x*c(3*x)/(1-x*c(3*x))), c(x) the g.f. of A000108.

1, 1, 1, 4, 5, 1, 25, 33, 9, 1, 190, 256, 78, 13, 1, 1606, 2186, 703, 139, 17, 1, 14506, 19863, 6591, 1430, 216, 21, 1, 137089, 188449, 63813, 14669, 2501, 309, 25, 1, 1338790, 1845416, 633808, 151532, 27940, 3980, 418, 29, 1, 13403950, 18513822, 6425196, 1580316, 307752, 48180, 5931, 543, 33, 1

Offset: 0

Author

Paul Barry, Jul 24 2005

Keywords

• easy
• nonn
• tabl

Comments

Product of (1, x*c(3*x)) and (1/(1-x), x/(1-x)) (A110518 and A007318). The binomial transform of the inverse of this triangle has general element (-3)^(n-k)*C(k,n-k), that is, it is the Riordan array (1, x*(1-3*x)) [A110517]. Row sums are A110520. Diagonal sums are A110521.

Examples

			Rows begin
     1;
     1,    1;
     4,    5,    1;
    25,   33,    9,    1;
   190,  256,   78,   13,    1;
  1606, 2186,  703,  139,   17,    1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Programs

Mathematica

T[0, 0] := 1; T[0, k_] := 0; T[n_, k_] := Sum[j*3^(n - j)*Binomial[2*n - j - 1, n - j]*Binomial[j, k]/n, {j, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

PARI

concat([1], for(n=1, 10, for(k=0,n, print1(sum(j=0,n, j*binomial(2*n-j-1,n-j)*binomial(j,k)*3^(n-j)/n), ", ")))) \\ G. C. Greubel, Aug 29 2017

Formula

Number triangle T(0,k) = 0^k, T(n,k) = Sum_{j=0..n} j*C(2n-j-1, n-j)* C(j, k)3^(n-j)/n, n > 0, k > 0. Deleham triangle Delta(0^n, 3-2*0^n) [see construction in A084938].