A000718 - OEIS

%I #42 Jun 11 2022 15:33:44

%S 1,2,6,20,65,226,883,3947,20089,115036,732171,5126901,39165917,

%T 324138010,2888934623,27587288507,281001801969,3041152133848,

%U 34849036364659,421526126267265,5367037330561365,71752003756908550

%N Boustrophedon transform of triangular numbers 1,1,3,6,10,...

%H Reinhard Zumkeller, <a href="/A000718/b000718.txt">Table of n, a(n) for n = 0..400</a>

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>

%H 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 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>.

%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>

%F E.g.f.: (sec(x) + tan(x))*(exp(x)*(x + 1/2*x^2) + 1). - _Sergei N. Gladkovskii_, Oct 30 2014

%F a(n) ~ n! * (8 + exp(Pi/2)*Pi*(4+Pi)) * 2^(n-1) / Pi^(n+1). - _Vaclav Kotesovec_, Jun 12 2015

%t 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 *)

%o (Haskell)

%o a000718 n = sum $ zipWith (*) (a109449_row n) (1 : tail a000217_list)

%o -- _Reinhard Zumkeller_, Nov 04 2013

%o (Python)

%o from itertools import accumulate, count, islice

%o def A000718_gen(): # generator of terms

%o     yield 1

%o     blist, c = (1,), 1

%o     for i in count(2):

%o         yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]

%o         c += i

%o A000718_list = list(islice(A000718_gen(),30)) # _Chai Wah Wu_, Jun 11 2022

%Y Cf. A109449, A000217, A000746.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_