A000736 Boustrophedon transform of Catalan numbers 1, 1, 1, 2, 5, 14, ...

1, 2, 4, 10, 32, 120, 513, 2455, 13040, 76440, 492231, 3465163, 26530503, 219754535, 1959181266, 18710532565, 190588702776, 2062664376064, 23636408157551, 285900639990875, 3640199365715769, 48665876423760247

Offset

0,2

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..400

Peter Luschny, An old operation on sequences: the Seidel transform

J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).

N. J. A. Sloane, Transforms

Wikipedia, Boustrophedon transform

Index entries for sequences related to boustrophedon transform

Formula

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

a(n) ~ n! * (6/Pi+2*exp(Pi)*((2-1/Pi)*BesselI(0,Pi)-2*BesselI(1,Pi))) * 2^n / Pi^n. - Vaclav Kotesovec, Oct 30 2014

Maple

egf := (sec(x/2)+tan(x/2))*(exp(x)*((x-1/2)*BesselI(0, x)-x*BesselI(1, x))+3/2);
s := n -> 2^n*n!*coeff(series(egf, x, n+2), x, n); seq(s(n), n=0..22); # Peter Luschny, Oct 30 2014, after Sergei N. Gladkovskii

Mathematica

CoefficientList[Series[1/2*(3 + E^(2*x)*((4*x-1)*BesselI[0, 2*x] - 4*x*BesselI[1, 2*x]))*(Sec[x] + Tan[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 30 2014, after Peter Luschny *)
t[n_, 0] := If[n == 0, 1, CatalanNumber[n - 1]]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

Prog

(Haskell)
a000736 n = sum $ zipWith (*) (a109449_row n) (1 : a000108_list)
-- Reinhard Zumkeller, Nov 05 2013
(Python)
from itertools import accumulate, count, islice
def A000736_gen(): # generator of terms
    yield 1
    blist, c = (1, ), 1
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist), initial=c)))[-1]
        c = c*(4*i+2)//(i+2)
A000736_list = list(islice(A000736_gen(), 40)) # Chai Wah Wu, Jun 12 2022

Crossrefs

Cf. A000108, A000753, A109449.

Sequence in context: A071954 A352279 A120017 * A263663 A176006 A263664

Adjacent sequences: A000733 A000734 A000735 * A000737 A000738 A000739

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Status

approved