A000737 Boustrophedon transform of natural numbers, cf. A000027.

1, 3, 8, 21, 60, 197, 756, 3367, 17136, 98153, 624804, 4375283, 33424512, 276622829, 2465449252, 23543304919, 239810132288, 2595353815825, 29740563986500, 359735190398875, 4580290700420064, 61233976084442741

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 44-54 1996 (Abstract, pdf, ps).
• N. J. A. Sloane, Transforms
• Wikipedia, Boustrophedon transform
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A231179.

Programs

Haskell

a000737 n = sum $ zipWith (*) (a109449_row n) [1..]
-- Reinhard Zumkeller, Nov 05 2013

Mathematica

CoefficientList[Series[(1+x)*(Tan[x]+1/Cos[x])* E^x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)
t[n_, 0] := 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 A000737_gen(): # generator of terms
    blist = tuple()
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A000737_list = list(islice(A000737_gen(),40)) # Chai Wah Wu, Jun 12 2022

Sage

# Algorithm of L. Seidel (1877)
def A000737_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = i+1
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        # To trace the algorithm remove the comment sign.
        # print([A[z] for z in (-i//2..i//2)])
        R.append(A[e*i//2])
    return R
A000737_list(10) # Peter Luschny, Jun 02 2012

Formula

E.g.f.: (1 + x)*(tan x + sec x)*exp(x).

a(n) ~ n! * (Pi + 2)*exp(Pi/2)*2^(n+1)/Pi^(n+1). - Vaclav Kotesovec, Oct 02 2013