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A109449
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T(n,k) = binomial(n,k)*A000111(n-k), triangle T(n,k) 0<=k<=n, read by rows.
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5
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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, 427, 336, 175, 70, 21, 7, 1, 1385, 2176, 1708, 896, 350, 112, 28, 8, 1, 7936, 12465, 9792, 5124, 2016, 630, 168, 36, 9, 1, 50521, 79360, 62325, 32640, 12810
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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.
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]
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LINKS
| 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 (Abstract, pdf, ps).
Peter Luschny, The Swiss-Knife polynomials. [From Peter Luschny (peter(AT)luschny. de), Jul 10 2009]
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FORMULA
| Sum_{k>=0} T(n, k) = A000667(n).
Sum_{k>=0} T(2n, 2k) = A000795(n).
Sum_{k>=0} T(2n, 2k+1) = A009747(n).
Sum_{k>=0} T(2n+1, 2k) = A003719(n).
Sum_{k>=0} T(2n+1, 2k+1) = A002084(n).
Sum_{k>=0} T(n, 2k) = A062272(n).
Sum_{k>=0} T(n, 2k+1) = A062161(n).
Sum_{k>=0}(-1)^(k)*T(n, k) = A062162(n). [From Johannes W. Meijer, Apr 20 2011]
E.g.f.: exp(x*y)*(sec(x)+tan(x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 20 2007
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]
Contribution from Reikku Kulon (reikku(AT)gmail.com), Feb 26 2009: (Start)
A109449(n, 0) = A000111(n), approx. round(2^(n + 2) * n! / Pi^(n + 1)).
A109449(n, n - 1) = n
A109449(n, n) = 1
For n > 0, k > 0:
A109449(n, k) = A109449(n - 1, k - 1) * n / k. (End)
Contribution from Peter Luschny (peter(AT)luschny.de), Jul 10 2009: (Start)
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. Another way to express
the polynomials p_n(x) is
p_n(x) = -x^n + sum_{k=0..n} C(n,k) Euler(k)((x+1)^(n-k)+x^(n-k)). (End)
An explicit formula for the n-th row polynomial is
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.
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. [Peter Bala]
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EXAMPLE
| Triangle starts:
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, 427, 336, 175, 70, 21, 7, 1;
1385, 2176, 1708, 896, 350, 112, 28, 8, 1;
7936, 12465, 9792, 5124, 2016, 630, 168, 36, 9, 1;
50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1; ...
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MAPLE
| Contribution from Peter Luschny (peter(AT)luschny.de), Jul 10 2009: (Start)
# Auxiliary functions
Pow := (n, k) -> `if`(n=0 and k=0, 1, n^k): # To avoid '0^0 undefined'.
Euler := (n, x) -> `if`(n=0, 1, euler(n, x)): # Avoid the bug euler(0, 1) = -1.
sigma := proc(n) local nmod8; nmod8 := n mod 8;
if n = 0 then RETURN(1) fi; if member(nmod8, {2, 6}) then RETURN(-1) fi;
if member(nmod8, {0, 1, 7}) then 1 else -1 fi; %*2^(-iquo(n-1, 2))-1 end:
A000111 := n -> 2^n*abs(Euler(n, 1/2)+Euler(n, 1))-`if`(n=0, 1, 0):
# Coefficients
A109449 := proc(n, k) binomial(n, k)*A000111(n-k) end:
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:
R109449 := proc(n, k) option remember; if k = 0 then RETURN(A000111(n)) fi; R109449(n-1, k-1)*n/k end:
# Polynomials
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:
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:
X109449 := proc(n) n!*coeff(series(exp(x*t)*(sech(t)+tanh(t)), t, 24), t, n)end:
# Evaluate
seq(print(seq(A109449(n, k), k=0..n)), n=0..9);
seq(print(seq(B109449(n, k), k=0..n)), n=0..9);
seq(print(seq(R109449(n, k), k=0..n)), n=0..9);
seq(print(seq(abs(coeff(E109449(n), x, k)), k=0..n)), n=0..9);
seq(print(seq(abs(coeff(L109449(n), x, k)), k=0..n)), n=0..9);
seq(print(seq(abs(coeff(X109449(n), x, k)), k=0..n)), n=0..9); (End)
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CROSSREFS
| Cf. A000111, A000667, A000795, A002084, A003719, A007318, A009747.
See also : A000182, A000964, A009739, A062161, A062272.
Cf. A153641, A162660. [From Peter Luschny (peter(AT)luschny.de), Jul 10 2009]
Sequence in context: A097724 A091836 A080850 * A129570 A165014 A058063
Adjacent sequences: A109446 A109447 A109448 * A109450 A109451 A109452
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KEYWORD
| nonn,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 27 2005
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EXTENSIONS
| 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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