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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

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

Crossrefs

Cf. A109449, A000079, A000752.

Programs

Haskell

a000734 n = sum $ zipWith (*) (a109449_row n) (1 : a000079_list)
-- Reinhard Zumkeller, Nov 04 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 *)

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

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

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