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, 593368400878431795, 8688251294851280594

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. A000111, A028310, A109449, A231179.

Programs

Haskell

a000660 n = sum $ zipWith (*) (a109449_row n) (1 : [1..])
-- Reinhard Zumkeller, Nov 04 2013

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 *)

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

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

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