A110506 Riordan array (1/(1-xc(2x)),xc(2x)/(1-xc(2x))), c(x) the g.f. of A000108.

1, 1, 1, 3, 4, 1, 13, 19, 7, 1, 67, 102, 44, 10, 1, 381, 593, 278, 78, 13, 1, 2307, 3640, 1795, 568, 121, 16, 1, 14589, 23231, 11849, 4051, 999, 173, 19, 1, 95235, 152650, 79750, 28770, 7820, 1598, 234, 22, 1, 636925, 1025965, 545680, 204760, 59650, 13642, 2392, 304, 25, 1

Offset

0,4

Comments

Deleham triangle Delta(0^n,2-0^n) [see construction in A084938]. The binomial transform of the inverse of this triangle has general element (-2)^(n-k)*C(k,n-k), that is, it is the Riordan array (1,x(1-2x)) [A110509]. Row sums are A052701. Diagonal sums are A110508. Inverse is A110511.

Links

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

Formula

T(0,0) = 1, T(n,k) = Sum_{j=0..n} j*C(2n-j-1,n-j) * C(j,k) * 2^(n-j))/n.

T(n,k) = (-1)^(n-k)*A114189(n,k). - Philippe Deléham, Mar 24 2007

Example

Rows begin:
1;
1,1;
3,4,1;
13,19,7,1;
67,102,44,10,1;
381,593,278,78,13,1;
From Philippe Deléham, Dec 01 2015: (Start)
Production matrix begins:
1, 1
2, 3, 1
2, 4, 3, 1
2, 4, 4, 3, 1
2, 4, 4, 4, 3, 1
2, 4, 4, 4, 4, 3, 1
2, 4, 4, 4, 4, 4, 3, 1
(End)

Mathematica

{{1}}~Join~Table[Sum[j Binomial[2 n - j - 1, n - j] Binomial[j, k] 2^(n - j), {j, 0, n}]/n, {n, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)

Prog

(PARI) tabl(nn)= {for (n=0, nn, for (k=0, n, if (n==0, x = 0^k, x = sum(j=0, n, j*binomial(2*n-j-1, n-j)*binomial(j, k)*2^(n-j)/n)); print1(x, ", "); ); print(); ); } \\ Michel Marcus, Jun 18 2015

Crossrefs

Cf. A000108, A052701, A084938, A110508, A110509, A110511, A114189.

Sequence in context: A109956 A123319 A076785 * A114189 A200659 A059110

Adjacent sequences: A110503 A110504 A110505 * A110507 A110508 A110509

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 24 2005

Status

approved