A112657 A Motzkin transform of Jacobsthal numbers.

1, 2, 7, 23, 79, 272, 943, 3278, 11419, 39830, 139057, 485795, 1697905, 5936348, 20760271, 72615143, 254028355, 888758030, 3109714117, 10881403229, 38077702909, 133251869648, 466325356273, 1631981113112, 5711490384901

Offset: 0

Author

Paul Barry, Jan 11 2006

Keywords

• easy
• nonn

Comments

Binomial transform of A100098.

Inverse binomial transform of A007854. The Hankel transform of this sequence is 3^n (see A000244). - Philippe Deléham, Nov 25 2007

Crossrefs

Cf. A000244, A007854, A026300, A089942, A100098.

Formula

a(n) = Sum_{k=0..n} A026300(n, k)*(2^(k+1) + (-1)^k)/3, where A026300 is the Motzkin triangle; a(n) = Sum_{k=0..n} ((k+1)/(n+1))*Sum_{j=0..n+1} C(n+1, j)*C(j, 2j-n+k)*(2^(k+1) + (-1)^k)/3.

a(n) = Sum_{k=0..n} A089942(n,k)*2^k = Sum_{k=0..n} A071947(n,k)*2^(n-k). - Philippe Deléham, Mar 31 2007