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

0,2

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

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

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 *)

Prog

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

Crossrefs

Cf. A231179.

Sequence in context: A273720 A018037 A018038 * A371891 A192235 A148769

Adjacent sequences: A000734 A000735 A000736 * A000738 A000739 A000740

Keyword

nonn

Author

N. J. A. Sloane

Status

approved