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

0,2

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

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

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 *)

Prog

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

Crossrefs

Cf. A000108, A000736, A109449.

Sequence in context: A185143 A280760 A350717 * A346813 A007878 A019589

Adjacent sequences: A000750 A000751 A000752 * A000754 A000755 A000756

Keyword

nonn

Author

N. J. A. Sloane

Status

approved