A260786 Twice the Euler or up/down numbers A000111.

2, 2, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042, 707584, 5405530, 44736512, 398721962, 3807514624, 38783024290, 419730685952, 4809759350882, 58177770225664, 740742376475050, 9902996106248192, 138697748786275802, 2030847773013704704, 31029068327114173810, 493842960380415967232

Offset: 0

Author

N. J. A. Sloane, Aug 04 2015

Keywords

• nonn

Links

• S. T. Thompson, Problem E754: Skew Ordered Sequences, Amer. Math. Monthly, 54 (1947), 416-417. [Annotated scanned copy]

Crossrefs

Cf. A000111.

Apart from initial terms, same as A001250.

Programs

Maple

f:=proc(n) option remember;
if n <= 1 then 2 else (1/4)*add(binomial(n-1,k-1)*f(k-1)*f(n-k),k=1..n); fi;
end;
[seq(f(n),n=0..30)];

Python

from itertools import accumulate, islice
def A260786_gen(): # generator of terms
    yield from (2,2)
    blist = (0,2)
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
A260786_list = list(islice(A260786_gen(),30)) # Chai Wah Wu, Apr 17 2023

Formula

a(0)=a(1)=2; thereafter a(n) = (1/4)*Sum_{k=1..n} binomial(n-1, k-1)*a(k-1)*a(n-k).