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

%I A230961 #19 Jun 12 2022 12:01:46
%S A230961 1,3,11,50,273,1746,12823,106462,986689,10103074,113309991,1381835454,
%T A230961 18209834849,257911743506,3907538236631,63066584719982,
%U A230961 1080340925760129,19577690297352258,374214932301757255,7524626434657416286,158783753482817132065
%N A230961 Boustrophedon transform of factorials beginning with 1, cf. A000142.
%H A230961 Reinhard Zumkeller, <a href="/A230961/b230961.txt">Table of n, a(n) for n = 0..400</a>
%H A230961 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%H A230961 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 A230961 Wikipedia, <a href="http://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon_transform</a>
%H A230961 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A230961 a(n) = sum(A109449(n,k)*A000142(k+1): k=0..n).
%F A230961 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 A230961 a(n) ~ n! * n * (1+sin(1))/cos(1). - _Vaclav Kotesovec_, Jun 12 2015
%t A230961 T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
%t A230961 a[n_] := Sum[T[n, k] (k + 1)!, {k, 0, n}];
%t A230961 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 23 2019 *)
%o A230961 (Haskell)
%o A230961 a230961 n = sum $ zipWith (*) (a109449_row n) $ tail a000142_list
%o A230961 (Python)
%o A230961 from itertools import accumulate, count, islice
%o A230961 def A230961_gen(): # generator of terms
%o A230961     blist, m = tuple(), 1
%o A230961     for i in count(1):
%o A230961         yield (blist := tuple(accumulate(reversed(blist),initial=(m := m*i))))[-1]
%o A230961 A230961_list = list(islice(A230961_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y A230961 Cf. A230960.
%K A230961 nonn
%O A230961 0,2
%A A230961 _Reinhard Zumkeller_, Nov 05 2013