A230961 - OEIS

%I #19 Jun 12 2022 12:01:46

%S 1,3,11,50,273,1746,12823,106462,986689,10103074,113309991,1381835454,

%T 18209834849,257911743506,3907538236631,63066584719982,

%U 1080340925760129,19577690297352258,374214932301757255,7524626434657416286,158783753482817132065

%N Boustrophedon transform of factorials beginning with 1, cf. A000142.

%H Reinhard Zumkeller, <a href="/A230961/b230961.txt">Table of n, a(n) for n = 0..400</a>

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>

%H 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 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon_transform</a>

%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>

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

%F 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

%F a(n) ~ n! * n * (1+sin(1))/cos(1). - _Vaclav Kotesovec_, Jun 12 2015

%t T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];

%t a[n_] := Sum[T[n, k] (k + 1)!, {k, 0, n}];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 23 2019 *)

%o (Haskell)

%o a230961 n = sum $ zipWith (*) (a109449_row n) $ tail a000142_list

%o (Python)

%o from itertools import accumulate, count, islice

%o def A230961_gen(): # generator of terms

%o     blist, m = tuple(), 1

%o     for i in count(1):

%o         yield (blist := tuple(accumulate(reversed(blist),initial=(m := m*i))))[-1]

%o A230961_list = list(islice(A230961_gen(),40)) # _Chai Wah Wu_, Jun 12 2022

%Y Cf. A230960.

%K nonn

%O 0,2

%A _Reinhard Zumkeller_, Nov 05 2013