A000734 Boustrophedon transform of 1,1,2,4,8,16,32,...

1, 2, 5, 15, 49, 177, 715, 3255, 16689, 95777, 609875, 4270695, 32624329, 269995377, 2406363835, 22979029335, 234062319969, 2533147494977, 29027730898595, 351112918079175, 4470508510495609, 59766296291090577

Offset

0,2

Comments

Binomial transform of A062272. - Paul Barry, Jan 21 2005

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 (1996) 44-54 (Abstract, pdf, ps).

N. J. A. Sloane, Transforms.

Wikipedia, Boustrophedon transform.

Index entries for sequences related to boustrophedon transform

Formula

E.g.f.: (1 + exp(2*x))*(sec(x) + tan(x))/2. - Paul Barry, Jan 21 2005

a(n) ~ n! * (1 + exp(Pi)) * (2/Pi)^(n+1). - Vaclav Kotesovec, Oct 07 2013

Mathematica

CoefficientList[Series[(1+E^(2*x))*(Sec[x]+Tan[x])/2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 07 2013 *)
t[n_, 0] := If[n == 0, 1, 2^(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

(Sage) # Algorithm of L. Seidel (1877)
def A000734_list(n) :
    A = {-1:0, 0:1}; R = []
    k = 0; e = 1; Bm = 1
    for i in range(n) :
        Am = Bm
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        Bm += Bm
        R.append(A[e*i//2]/2)
    return R
A000734_list(22) # Peter Luschny, Jun 02 2012
(Haskell)
a000734 n = sum $ zipWith (*) (a109449_row n) (1 : a000079_list)
-- Reinhard Zumkeller, Nov 04 2013
(Python)
from itertools import count, accumulate, islice
def A000734_gen(): # generator of terms
    yield 1
    blist, m = (1, ), 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist), initial=m)))[-1]
        m *= 2
A000734_list = list(islice(A000734_gen(), 40)) # Chai Wah Wu, Jun 12 2022

Crossrefs

Cf. A109449, A000079, A000752.

Sequence in context: A149942 A149943 A079146 * A148366 A005751 A202182

Adjacent sequences: A000731 A000732 A000733 * A000735 A000736 A000737

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Status

approved