A000752 Boustrophedon transform of powers of 2.

1, 3, 9, 28, 93, 338, 1369, 6238, 31993, 183618, 1169229, 8187598, 62545893, 517622498, 4613366689, 44054301358, 448733127793, 4856429646978, 55650582121749, 673136951045518, 8570645832753693, 114581094529057058, 1604780986816602409, 23497612049668468078

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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. A000079, A000734.

Column k=2 of A292975.

Programs

Haskell

a000752 n = sum $ zipWith (*) (a109449_row n) a000079_list
-- Reinhard Zumkeller, Nov 03 2013

Mathematica

t[n_, 0] := 2^n; 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 *)
With[{nn=30},CoefficientList[Series[Exp[2x](Tan[ x]+Sec[x]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 15 2018 *)

Python

from itertools import accumulate, islice
def A000752_gen(): # generator of terms
    blist, m = tuple(), 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m *= 2
A000752_list = list(islice(A000752_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

E.g.f.: exp(2*x) (tan(x) + sec(x)).

a(n) = Sum_{k=0..n} A109449(n,k)*2^k. - Reinhard Zumkeller, Nov 03 2013

G.f.: E(0)*x/(1 - 2*x)/(1 - 3*x) + 1/(1 - 2*x), where E(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(x*(k+3) - 1)*(x*(k+4) -1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2014

a(n) ~ n! * exp(Pi) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015