A000718 Boustrophedon transform of triangular numbers 1,1,3,6,10,...

1, 2, 6, 20, 65, 226, 883, 3947, 20089, 115036, 732171, 5126901, 39165917, 324138010, 2888934623, 27587288507, 281001801969, 3041152133848, 34849036364659, 421526126267265, 5367037330561365, 71752003756908550

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

a(n) ~ n! * (8 + exp(Pi/2)*Pi*(4+Pi)) * 2^(n-1) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

Mathematica

t[n_, 0] := If[n == 0, 1, n*(n+1)/2]; 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

(Haskell)
a000718 n = sum $ zipWith (*) (a109449_row n) (1 : tail a000217_list)
-- Reinhard Zumkeller, Nov 04 2013
(Python)
from itertools import accumulate, count, islice
def A000718_gen(): # generator of terms
    yield 1
    blist, c = (1, ), 1
    for i in count(2):
        yield (blist := tuple(accumulate(reversed(blist), initial=c)))[-1]
        c += i
A000718_list = list(islice(A000718_gen(), 30)) # Chai Wah Wu, Jun 11 2022

Crossrefs

Cf. A109449, A000217, A000746.

Sequence in context: A302612 A005726 A148473 * A148474 A156831 A027061

Adjacent sequences: A000715 A000716 A000717 * A000719 A000720 A000721

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Status

approved