A110519 Riordan array (1/(1-xc(3x)), xc(3x)/(1-xc(3x))), 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

Offset

0,4

Comments

Product of (1, xc(3x)) 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(1-3x)) [A110517]. Row sums are A110520. Diagonal sums are A110521.

Links

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

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].

Example

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

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, 20}, {k, 0, n}] // Flatten ( G. C. Greubel, Aug 29 2017 *)

Prog

(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

Crossrefs

Sequence in context: A109962 A102230 A147724 * A286796 A286718 A204579

Adjacent sequences: A110516 A110517 A110518 * A110520 A110521 A110522

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 24 2005

Status

approved