A230961 Boustrophedon transform of factorials beginning with 1, cf. A000142.

1, 3, 11, 50, 273, 1746, 12823, 106462, 986689, 10103074, 113309991, 1381835454, 18209834849, 257911743506, 3907538236631, 63066584719982, 1080340925760129, 19577690297352258, 374214932301757255, 7524626434657416286, 158783753482817132065

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. A230960.

Programs

Haskell

a230961 n = sum $ zipWith (*) (a109449_row n) $ tail 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 + 1)!, {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 23 2019 *)

Python

from itertools import accumulate, count, islice
def A230961_gen(): # generator of terms
    blist, m = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=(m := m*i))))[-1]
A230961_list = list(islice(A230961_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

a(n) = sum(A109449(n,k)*A000142(k+1): k=0..n).

E.g.f.: conjecture: (tan(x)+sec(x))/(1-2*x+x^2) = (1- 12*x/ (Q(0)+6*x+3*x^2))/(1-x)^2, 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! * n * (1+sin(1))/cos(1). - Vaclav Kotesovec, Jun 12 2015