A230960 Boustrophedon transform of factorials, cf. A000142.

1, 2, 5, 17, 73, 381, 2347, 16701, 134993, 1222873, 12279251, 135425553, 1627809401, 21183890469, 296773827547, 4453511170517, 71275570240417, 1211894559430065, 21816506949416611, 414542720924028441, 8291224789668806345, 174120672081098057341

Offset: 0

Author

Reinhard Zumkeller, Nov 05 2013

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).
• Wikipedia, Boustrophedon_transform
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000142, A092580, A109449, A230961.

Programs

Haskell

a230960 n = sum $ zipWith (*) (a109449_row n) a000142_list

Mathematica

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
a[n_] := Sum[T[n, k] k!, {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 02 2019 *)

Python

from itertools import count, islice, accumulate
def A230960_gen(): # generator of terms
    blist, m = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m *= i
A230960_list = list(islice(A230960_gen(),30)) # Chai Wah Wu, Jun 11 2022

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*A000142(k).

E.g.f.: (tan(x)+sec(x))/(1-x) = (1- 12*x/(Q(0)+6*x+3*x^2))/(1-x), where Q(k) = 2*(4*k+1)*(32*k^2+16*k-x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013

a(n) ~ n! * (1+sin(1))/cos(1). - Vaclav Kotesovec, Jun 12 2015

a(n) = Sum_{k=0..n} (k+1) * A092580(n,k). - Alois P. Heinz, Apr 27 2023