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%I #72 Jun 06 2022 09:13:29
%S 1,1,1,1,2,1,2,3,3,1,5,8,6,4,1,16,25,20,10,5,1,61,96,75,40,15,6,1,272,
%T 427,336,175,70,21,7,1,1385,2176,1708,896,350,112,28,8,1,7936,12465,
%U 9792,5124,2016,630,168,36,9,1,50521,79360,62325,32640,12810,4032,1050,240,45,10,1
%N Triangle read by rows, T(n,k) = binomial(n,k)*A000111(n-k), 0 <= k <= n.
%C The boustrophedon transform {t} of a sequence {s} is given by t_n = Sum_{k=0..n} T(n,k)*s(k). Triangle may be called the boustrophedon triangle.
%C The 'signed version' of the triangle is the exponential Riordan array [sech(x)+tanh(x), x]. - _Peter Luschny_, Jan 24 2009
%C Up to signs, the matrix is self-inverse: T^(-1)(n,k) = (-1)^(n+k)*T(n,k). - _R. J. Mathar_, Mar 15 2013
%H Reinhard Zumkeller, <a href="/A109449/b109449.txt">Rows n = 0..125 of table, flattened</a>
%H Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html"> The Swiss-Knife polynomials.</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="https://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 Sum_{k>=0} T(n, k) = A000667(n).
%F Sum_{k>=0} T(2n, 2k) = A000795(n).
%F Sum_{k>=0} T(2n, 2k+1) = A009747(n).
%F Sum_{k>=0} T(2n+1, 2k) = A003719(n).
%F Sum_{k>=0} T(2n+1, 2k+1) = A002084(n).
%F Sum_{k>=0} T(n, 2k) = A062272(n).
%F Sum_{k>=0} T(n, 2k+1) = A062161(n).
%F Sum_{k>=0} (-1)^(k)*T(n, k) = A062162(n). - _Johannes W. Meijer_, Apr 20 2011
%F E.g.f.: exp(x*y)*(sec(x)+tan(x)). - _Vladeta Jovovic_, May 20 2007
%F T(n,k) = 2^(n-k)C(n,k)|E(n-k,1/2)+E(n-k,1)|-[n=k] where C(n,k) is the binomial coefficient, E(m,x) are the Euler polynomials and [] the Iverson bracket. - _Peter Luschny_, Jan 24 2009
%F From _Reikku Kulon_, Feb 26 2009: (Start)
%F A109449(n, 0) = A000111(n), approx. round(2^(n + 2) * n! / Pi^(n + 1)).
%F A109449(n, n - 1) = n.
%F A109449(n, n) = 1.
%F For n > 0, k > 0: A109449(n, k) = A109449(n - 1, k - 1) * n / k. (End)
%F From _Peter Luschny_, Jul 10 2009: (Start)
%F Let p_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v C(k,v)F(k)(x+v+1)^n, where F(0)=1 and for k>0 F(k)=-1 + s_k 2^floor((k-1)/2), s_k is 0 if k mod 8 in {2,6}, 1 if k mod 8 in {0,1,7} and otherwise -1. T(n,k) are the absolute values of the coefficients of these polynomials.
%F Another way to express the polynomials p_n(x) is
%F p_n(x) = -x^n + Sum_{k=0..n} binomial(n,k) Euler(k)((x+1)^(n-k)+x^(n-k)). (End)
%F From _Peter Bala_, Jan 26 2011: (Start)
%F An explicit formula for the n-th row polynomial is
%F x^n + i*Sum_{k=1..n}((1+i)/2)^(k-1)*Sum_{j=0..k} (-1)^j*binomial(k,j)*(x+i*j)^n, where i = sqrt(-1). This is the triangle of connection constants between the polynomial sequences {Z(n,x+1)} and {Z(n,x)}, where Z(n,x) denotes the zigzag polynomials described in A147309.
%F Denote the present array by M. The first column of the array (I-x*M)^-1 is a sequence of rational functions in x whose numerator polynomials are the row polynomials of A145876 - the generalized Eulerian numbers associated with the zigzag numbers. (End)
%F Let skp{n}(x) denote the Swiss-Knife polynomials A153641. Then
%F T(n,k) = [x^(n-k)] |skp{n}(x) - skp{n}(x-1) + x^n|. - _Peter Luschny_, Jul 22 2012
%F T(n,k) = A007318(n,k) * A000111(n - k), k = 0..n. - _Reinhard Zumkeller_, Nov 02 2013
%F T(n,k) = abs(A247453(n,k)). - _Reinhard Zumkeller_, Sep 17 2014
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 2, 3, 3, 1;
%e 5, 8, 6, 4, 1;
%e 16, 25, 20, 10, 5, 1;
%e 61, 96, 75, 40, 15, 6, 1;
%e 272, 427, 336, 175, 70, 21, 7, 1;
%e 1385, 2176, 1708, 896, 350, 112, 28, 8, 1;
%e 7936, 12465, 9792, 5124, 2016, 630, 168, 36, 9, 1;
%e 50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1; ...
%p From _Peter Luschny_, Jul 10 2009, edited Jun 06 2022: (Start)
%p A109449 := (n,k) -> binomial(n, k)*A000111(n-k):
%p seq(print(seq(A109449(n, k), k=0..n)), n=0..9);
%p B109449 := (n,k) -> 2^(n-k)*binomial(n, k)*abs(euler(n-k, 1/2)+euler(n-k, 1)) -`if`(n-k=0, 1, 0): seq(print(seq(B109449(n, k), k=0..n)), n=0..9);
%p R109449 := proc(n, k) option remember; if k = 0 then A000111(n) else R109449(n-1, k-1)*n/k fi end: seq(print(seq(R109449(n, k), k=0..n)), n=0..9);
%p E109449 := proc(n) add(binomial(n, k)*euler(k)*((x+1)^(n-k)+ x^(n-k)), k=0..n) -x^n end: seq(print(seq(abs(coeff(E109449(n), x, k)), k=0..n)), n=0..9);
%p sigma := n -> ifelse(n=0, 1, [1,1,0,-1,-1,-1,0,1][n mod 8 + 1]/2^iquo(n-1,2)-1):
%p L109449 := proc(n) add(add((-1)^v*binomial(k, v)*(x+v+1)^n*sigma(k), v=0..k), k=0..n) end: seq(print(seq(abs(coeff(L109449(n), x, k)), k=0..n)), n=0..9);
%p X109449 := n -> n!*coeff(series(exp(x*t)*(sech(t)+tanh(t)), t, 24), t, n): seq(print(seq(abs(coeff(X109449(n), x, k)), k=0..n)), n=0..9);
%p (End)
%t lim = 10; s = CoefficientList[Series[(1 + Sin[x])/Cos[x], {x, 0, lim}], x] Table[k!, {k, 0, lim}]; Table[Binomial[n, k] s[[n - k + 1]], {n, 0, lim}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 24 2015, after _Jean-François Alcover_ at A000111 *)
%t T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 27 2019 *)
%o (Sage)
%o R = PolynomialRing(ZZ, 'x')
%o @CachedFunction
%o def skp(n, x) :
%o if n == 0 : return 1
%o return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
%o def A109449_row(n):
%o x = R.gen()
%o return [abs(c) for c in list(skp(n,x)-skp(n,x-1)+x^n)]
%o for n in (0..10) : print(A109449_row(n)) # _Peter Luschny_, Jul 22 2012
%o (Haskell)
%o a109449 n k = a109449_row n !! k
%o a109449_row n = zipWith (*)
%o (a007318_row n) (reverse $ take (n + 1) a000111_list)
%o a109449_tabl = map a109449_row [0..]
%o -- _Reinhard Zumkeller_, Nov 02 2013
%o (PARI) A109449(n,k)=binomial(n,k)*if(n>k,2*abs(polylog(k-n,I)),1) \\ _M. F. Hasler_, Oct 05 2017
%Y Cf. A000111, A000667, A000795, A002084, A003719, A007318, A009747.
%Y See also : A000182, A000964, A009739, A062161, A062272.
%Y Cf. A153641, A162660.
%Y Cf. A000667 (row sums), A247453.
%K nonn,tabl
%O 0,5
%A _Philippe Deléham_, Aug 27 2005
%E Edited, formula corrected, typo T(9,4)=2016 (before 2816) fixed by _Peter Luschny_, Jul 10 2009
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