A000754 Boustrophedon transform of odd numbers.

1, 4, 12, 33, 96, 317, 1218, 5425, 27608, 158129, 1006574, 7048657, 53847420, 445643681, 3971876930, 37928628529, 386337833232, 4181155148673, 47912508680086, 579538956964241, 7378919177090244, 98648882783190305

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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. A005408.

Programs

Haskell

a000754 n = sum $ zipWith (*) (a109449_row n) [1, 3 ..]
-- Reinhard Zumkeller, Nov 02 2013

Mathematica

CoefficientList[Series[(Sec[x]+Tan[x])*E^x*(2*x+1), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 30 2014 after Sergei N. Gladkovskii *)
t[n_, 0] := 2n + 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 accumulate, count, islice
def A000754_gen(): # generator of terms
    blist = tuple()
    for i in count(1,2):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A000754_list = list(islice(A000754_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

From Reinhard Zumkeller, Nov 02 2013: (Start)

a(n) = Sum_{k=0..n} A116666(n+1,k)*A000111(n-k).

a(n) = Sum_{k=0..n} A109449(n,k)*(2*k + 1). (End)

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

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