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A110519 Riordan array (1/(1-xc(3x)), xc(3x)/(1-xc(3x))), c(x) the g.f. of A000108. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified February 22 13:09 EST 2018. Contains 299454 sequences. (Running on oeis4.)