cp's OEIS Frontend

A000746 Boustrophedon transform of triangular numbers.

%I A000746 #49 Jul 11 2025 01:13:55
%S A000746 1,4,13,39,120,407,1578,7042,35840,205253,1306454,9148392,69887664,
%T A000746 578392583,5155022894,49226836114,501420422112,5426640606697,
%U A000746 62184720675718,752172431553308,9576956842743904,128034481788227195
%N A000746 Boustrophedon transform of triangular numbers.
%H A000746 Reinhard Zumkeller, <a href="/A000746/b000746.txt">Table of n, a(n) for n = 0..400</a>
%H A000746 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.
%H A000746 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996), 44-54 (<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 A000746 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A000746 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>.
%H A000746 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A000746 a(n) = Sum_{k=0..n} A109449(n,k)*(k + 1)*(k + 2)/2. - _Reinhard Zumkeller_, Nov 03 2013
%F A000746 E.g.f.: (sec(x) + tan(x))*exp(x)*(x^2 + 4*x + 2)/2. - _Sergei N. Gladkovskii_, Oct 30 2014
%F A000746 a(n) ~ n! * (Pi^2 + 8*Pi + 8) * exp(Pi/2) * 2^(n-1) / Pi^(n+1). - _Vaclav Kotesovec_, Jun 12 2015
%t A000746 t[n_, 0] := (n + 1) (n + 2)/2; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 30, 0] (* _Jean-François Alcover_, Feb 12 2016 *)
%o A000746 (Haskell)
%o A000746 a000746 n = sum $ zipWith (*) (a109449_row n) $ tail a000217_list
%o A000746 -- _Reinhard Zumkeller_, Nov 03 2013
%o A000746 (Python)
%o A000746 from itertools import accumulate, count, islice
%o A000746 def A000746_gen(): # generator of terms
%o A000746     blist, c = tuple(), 1
%o A000746     for i in count(2):
%o A000746         yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
%o A000746         c += i
%o A000746 A000746_list = list(islice(A000746_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%o A000746 (Magma)
%o A000746 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A000746 Coefficients(R!(Laplace( (Sec(x)+Tan(x))*Exp(x)*(x^2+4*x+2)/2 ))); // _G. C. Greubel_, Jul 10 2025
%o A000746 (SageMath)
%o A000746 @CachedFunction
%o A000746 def f(n, k):
%o A000746     if (k==0): return binomial(n+2,2)
%o A000746     else: return f(n, k-1) + f(n-1, n-k)
%o A000746 def A000746(n): return f(n,n)
%o A000746 [A000746(n) for n in range(31)] # _G. C. Greubel_, Jul 10 2025
%Y A000746 Cf. A000217, A000718, A109449.
%K A000746 nonn
%O A000746 0,2
%A A000746 _N. J. A. Sloane_