A000697 Boustrophedon transform of 1, 1, 4, 9, 16, ...

1, 2, 7, 26, 89, 316, 1243, 5564, 28321, 162160, 1032051, 7226636, 55206161, 456886912, 4072080587, 38885496092, 396084390849, 4286637591872, 49121248360291, 594159600856332, 7565074996215025, 101137602761945440

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

Crossrefs

Cf. A000290, A000745, A109449.

Programs

Haskell

a000697 n = sum $ zipWith (*) (a109449_row n) (1 : tail a000290_list)
-- Reinhard Zumkeller, Nov 04 2013

Mathematica

t[n_, 0] := If[n==0, 1, n^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 A000697_gen(): # generator of terms
    yield 1
    blist, m = (1,), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m += 2*i+1
A000697_list = list(islice(A000697_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

E.g.f.: (1 + exp(x)*x*(1 + x))*(sec(x) + tan(x)). - Sergei N. Gladkovskii, Oct 29 2014

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