OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..517
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2.
FORMULA
a(n) = Sum_{k=0..n-1} binomial(n+k-1, n-k-1)*a(k) for n > 0 with a(0) = 1.
G.f. satisfies: A(x) = 1 + x*A(x/(1-x)^2) / (1-x). - Paul D. Hanna, Aug 15 2007
EXAMPLE
a(3) = 1*(1) + 3*(1) + 1*(2) = 6;
a(4) = 1*(1) + 6*(1) + 5*(2) + 1*(6) = 23;
a(5) = 1*(1) + 10*(1) + 15*(2) + 7*(6) + 1*(23) = 106.
Triangle A085478(n,k) = binomial(n+k, n-k) (with rows n >= 0 and columns k = 0..n) begins:
1;
1, 1;
1, 3, 1;
1, 6, 5, 1;
1, 10, 15, 7, 1;
1, 15, 35, 28, 9, 1;
...
where g.f. of column k = 1/(1-x)^(2*k+1).
MATHEMATICA
PROG
(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, a(k)*binomial(n+k-1, n-k-1)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 26 2006
STATUS
approved