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

Author

N. J. A. Sloane, Simon Plouffe

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. A109449, A000217, A000746.

Programs

Haskell

a000718 n = sum $ zipWith (*) (a109449_row n) (1 : tail a000217_list)
-- Reinhard Zumkeller, Nov 04 2013

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 *)

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

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