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
Examples
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)
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
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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 *) -
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
Formula
T(0,0) = 1, T(n,k) = (Sum_{j=0..n} j*C(2*n-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
Comments