Search: id:A109449
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%I A109449
%S A109449 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,
%T A109449 272,427,336,175,70,21,7,1,1385,2176,1708,896,350,112,28,8,1,7936,
%U A109449 12465,9792,5124,2016,630,168,36,9,1,50521,79360,62325,32640,12810
%N A109449 T(n,k) = binomial(n,k)*A000111(n-k), triangle T(n,k) 0<=k<=n, read by 
               rows.
%C A109449 The boustrophedon transform {t} of a sequence {s} is given by t_n = Sum_{k, 
               0<=k<=n} T(n,k)*s(k). Triangle may be called the boustrophedon triangle.
%C A109449 The 'signed version' of the triangle is the exponential Riordan array 
               [sech(x)+tanh(x), x]. [From Peter Luschny (peter(AT)luschny.de), 
               Jan 24 2009]
%H A109449 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: 
               the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 
               (<a href="http://www.research.att.com/~njas/doc/bous.txt">Abstract</
               a>, <a href="http://www.research.att.com/~njas/doc/bous.pdf">pdf</
               a>, <a href="http://www.research.att.com/~njas/doc/bous.ps">ps</a>
               ).
%H A109449 Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html"> 
               The Swiss-Knife polynomials.</a> [From Peter Luschny (peter(AT)luschny. 
               de), Jul 10 2009]
%F A109449 Sum_{k>=0} T(n, k) = A000667(n).
%F A109449 Sum_{k>=0} T(2n, 2k) = A000795(n).
%F A109449 Sum_{k>=0} T(2n, 2k+1) = A009747(n).
%F A109449 Sum_{k>=0} T(2n+1, 2k) = A003719(n).
%F A109449 Sum_{k>=0} T(2n+1, 2k+1) = A002084(n).
%F A109449 Sum_{k>=0} T(n, 2k) = A062272(n).
%F A109449 Sum_{k>=0} T(n, 2k+1) = A062161(n).
%F A109449 E.g.f.: exp(x*y)*(sec(x)+tan(x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               May 20 2007
%F A109449 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. [From Peter Luschny (peter(AT)luschny.de), Jan 24 
               2009]
%F A109449 Contribution from Reikku Kulon (reikku(AT)gmail.com), Feb 26 2009: (Start)
%F A109449 A109449(n, 0) = A000111(n), approx. round(2^(n + 2) * n! / Pi^(n + 1)).
%F A109449 A109449(n, n - 1) = n
%F A109449 A109449(n, n) = 1
%F A109449 For n > 0, k > 0:
%F A109449 A109449(n, k) = A109449(n - 1, k - 1) * n / k. (End)
%F A109449 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 10 2009: (Start)
%F A109449 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 A109449 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
%F A109449 {2,6}, 1 if k mod 8 in {0,1,7} and otherwise -1. T(n,k) are the absolute
%F A109449 values of the coefficients of these polynomials. Another way to express
%F A109449 the polynomials p_n(x) is
%F A109449 p_n(x) = -x^n + sum_{k=0..n} C(n,k) Euler(k)((x+1)^(n-k)+x^(n-k)). (End)
%e A109449 Triangle starts:
%e A109449 1;
%e A109449 1, 1;
%e A109449 1, 2, 1;
%e A109449 2, 3, 3, 1;
%e A109449 5, 8, 6, 4, 1;
%e A109449 16, 25, 20, 10, 5, 1;
%e A109449 61, 96, 75, 40, 15, 6, 1;
%e A109449 272, 427, 336, 175, 70, 21, 7, 1;
%e A109449 1385, 2176, 1708, 896, 350, 112, 28, 8, 1;
%e A109449 7936, 12465, 9792, 5124, 2016, 630, 168, 36, 9, 1;
%e A109449 50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1; ...
%p A109449 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 10 2009: (Start)
%p A109449 # Auxiliary functions
%p A109449 Pow := (n,k) -> `if`(n=0 and k=0,1,n^k): # To avoid '0^0 undefined'.
%p A109449 Euler := (n,x) -> `if`(n=0,1,euler(n,x)): # Avoid the bug euler(0,1) 
               = -1.
%p A109449 sigma := proc(n) local nmod8; nmod8 := n mod 8;
%p A109449 if n = 0 then RETURN(1) fi; if member(nmod8,{2,6}) then RETURN(-1) fi;
%p A109449 if member(nmod8,{0,1,7}) then 1 else -1 fi; %*2^(-iquo(n-1,2))-1 end:
%p A109449 A000111 := n -> 2^n*abs(Euler(n,1/2)+Euler(n,1))-`if`(n=0,1,0):
%p A109449 # Coefficients
%p A109449 A109449 := proc(n,k) binomial(n,k)*A000111(n-k) end:
%p A109449 B109449 := proc(n,k) 2^(n-k)*binomial(n,k)*abs(Euler(n-k,1/2)+Euler(n-k,
               1)) -`if`(n-k=0,1,0) end:
%p A109449 R109449 := proc(n,k) option remember; if k = 0 then RETURN(A000111(n)) 
               fi; R109449(n-1,k-1)*n/k end:
%p A109449 # Polynomials
%p A109449 E109449 := proc(n) local k; add(binomial(n,k)*euler(k)*(Pow(x+1,n-k)+ 
               Pow(x,n-k)),k=0..n)-Pow(x,n) end:
%p A109449 L109449 := proc(n) local k,v; add(add((-1)^v*binomial(k,v)*Pow(x+v+1,
               n)* sigma(k),v=0..k),k=0..n) end:
%p A109449 X109449 := proc(n) n!*coeff(series(exp(x*t)*(sech(t)+tanh(t)),t,24),t,
               n)end:
%p A109449 # Evaluate
%p A109449 seq(print(seq(A109449(n,k),k=0..n)),n=0..9);
%p A109449 seq(print(seq(B109449(n,k),k=0..n)),n=0..9);
%p A109449 seq(print(seq(R109449(n,k),k=0..n)),n=0..9);
%p A109449 seq(print(seq(abs(coeff(E109449(n),x,k)),k=0..n)),n=0..9);
%p A109449 seq(print(seq(abs(coeff(L109449(n),x,k)),k=0..n)),n=0..9);
%p A109449 seq(print(seq(abs(coeff(X109449(n),x,k)),k=0..n)),n=0..9); (End)
%Y A109449 Cf. A000111, A000667, A000795, A002084, A003719, A007318, A009747.
%Y A109449 See also : A000182, A000964, A009739, A062161, A062272.
%Y A109449 Cf. A153641, A162660. [From Peter Luschny (peter(AT)luschny.de), Jul 
               10 2009]
%Y A109449 Sequence in context: A097724 A091836 A080850 this_sequence A129570 A165014 
               A058063
%Y A109449 Adjacent sequences: A109446 A109447 A109448 this_sequence A109450 A109451 
               A109452
%K A109449 nonn,tabl
%O A109449 0,5
%A A109449 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 27 2005
%E A109449 Edited, formula corrected, typo T(9,4)=2016 (before 2816) fixed by Peter 
               Luschny (peter(AT)luschny.de), Jul 10 2009

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