A000660 Boustrophedon transform of 1,1,2,3,4,5,...

1, 2, 5, 14, 41, 136, 523, 2330, 11857, 67912, 432291, 3027166, 23125673, 191389108, 1705788659, 16289080922, 165919213089, 1795666675824, 20576824369027, 248892651678198, 3168999664907705, 42366404751871660

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

a(n) = Sum_{k=0..n} A109449(n,k)*A028310(k). - Reinhard Zumkeller, Nov 04 2013

E.g.f.: (x*exp(x) + 1)*(sec(x) + tan(x)). - Sergei N. Gladkovskii, Oct 28 2014

a(n) = A231179(n) + A000111(n). - Sergei N. Gladkovskii, Oct 28 2014

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

Maple

seq(coeff(series(factorial(n)*(x*exp(x)+1)*(sec(x)+tan(x)), x, n+1), x, n), n=0..25); # Muniru A Asiru, Jul 30 2018

Mathematica

a[n_] := n! SeriesCoefficient[(1+x Exp[x])(1+Sin[x])/Cos[x], {x, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 30 2018, after Sergei N. Gladkovskii *)

Prog

(Sage) # Algorithm of L. Seidel (1877)
def A000660_list(n) :
    R = []; A = {-1:0, 0:1}
    k = 0; e = 1
    for i in range(n) :
        Am = i
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        print([A[z] for z in (-i//2..i//2)])
        R.append(A[e*i//2])
    return R
A000660_list(10) # Peter Luschny, Jun 02 2012
(Haskell)
a000660 n = sum $ zipWith (*) (a109449_row n) (1 : [1..])
-- Reinhard Zumkeller, Nov 04 2013
(Python)
from itertools import accumulate, count, islice
def A000660_gen(): # generator of terms
    yield 1
    blist = (1, )
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist), initial=i)))[-1]
A000660_list = list(islice(A000660_gen(), 40)) # Chai Wah Wu, Jun 12 2022

Crossrefs

Sequence in context: A148322 A148323 A148324 * A148325 A281594 A174176

Adjacent sequences: A000657 A000658 A000659 * A000661 A000662 A000663

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Status

approved