A000753 Boustrophedon transform of Catalan numbers.

1, 2, 5, 16, 59, 243, 1101, 5461, 29619, 175641, 1137741, 8031838, 61569345, 510230087, 4549650423, 43452408496, 442620720531, 4790322653809, 54893121512453, 663974736739232, 8453986695437957, 113021461431438475

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..400
• J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996), 44-54 (Abstract, pdf, ps).
• N. J. A. Sloane, Transforms.
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000108, A000736, A109449.

Programs

Haskell

a000753 n = sum $ zipWith (*) (a109449_row n) a000108_list
-- Reinhard Zumkeller, Nov 05 2013

Mathematica

CoefficientList[Series[E^(2*x) * (BesselI[0,2*x] - BesselI[1,2*x]) * (Sec[x] + Tan[x]),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Oct 30 2014 after Sergei N. Gladkovskii *)

Python

from itertools import accumulate, count, islice
def A000753_gen(): # generator of terms
    blist, c = tuple(), 1
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
        c = c*(4*i+2)//(i+2)
A000753_list = list(islice(A000753_gen(),30)) # Chai Wah Wu, Jun 11 2022

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*A000108(k). - Reinhard Zumkeller, Nov 05 2013

E.g.f.: (sec(x) + tan(x))*exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)). - Sergei N. Gladkovskii, Oct 30 2014

a(n) ~ n! * exp(Pi) * (BesselI(0, Pi) - BesselI(1, Pi)) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Oct 30 2014