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A147309 Riordan array [sec(x), log(sec(x) + tan(x))].

%I A147309 #57 Mar 14 2020 17:08:08
%S A147309 1,0,1,1,0,1,0,4,0,1,5,0,10,0,1,0,40,0,20,0,1,61,0,175,0,35,0,1,0,768,
%T A147309 0,560,0,56,0,1,1385,0,4996,0,1470,0,84,0,1,0,24320,0,22720,0,3360,0,
%U A147309 120,0,1
%N A147309 Riordan array [sec(x), log(sec(x) + tan(x))].
%C A147309 Production array is [cosh(x),x] beheaded. Inverse is A147308. Row sums are A000111(n+1).
%C A147309 Unsigned version of A147308. - _N. J. A. Sloane_, Nov 07 2008
%C A147309 From _Peter Bala_, Jan 26 2011: (Start)
%C A147309 Define a polynomial sequence {Z(n,x)} n >= 0 by means of the recursion
%C A147309 (1)... Z(n+1,x) = 1/2*x*{Z(n,x-1)+Z(n,x+1)}
%C A147309 with starting condition Z(0,x) = 1. We call Z(n,x) the zigzag polynomial of degree n. This table lists the coefficients of these polynomials (for n >= 1) in ascending powers of x, row indices shifted by 1. The first few polynomials are
%C A147309 ... Z(1,x) = x
%C A147309 ... Z(2,x) = x^2
%C A147309 ... Z(3,x) = x + x^3
%C A147309 ... Z(4,x) = 4*x^2 + x^4
%C A147309 ... Z(5,x) = 5*x + 10*x^3 + x^5.
%C A147309 The value Z(n,1) equals the zigzag number A000111(n). The polynomials Z(n,x) occur in formulas for the enumeration of permutations by alternating descents A145876 and in the enumeration of forests of non-plane unary binary labeled trees A147315.
%C A147309 {Z(n,x)}n>=0 is a polynomial sequence of binomial type and so is analogous to the sequence of monomials x^n. Denoting Z(n,x) by x^[n] to emphasize this analogy, we have, for example, the following analog of Bernoulli's formula for the sum of integer powers:
%C A147309 (2)... 1^[m]+...+(n-1)^[m] = (1/(m+1))*Sum_{k=0..m} (-1)^floor(k/2)*binomial(m+1,k)*B_k*n^[m+1-k],
%C A147309 where {B_k} k >= 0 = [1, -1/2, 1/6, 0, -1/30, ...] is the sequence of Bernoulli numbers.
%C A147309 For similarly defined polynomial sequences to Z(n,x) see A185415, A185417 and A185419. See also A185424.
%C A147309 (End)
%C A147309 [gd(x)^(-1)]^m = Sum_{n>=m} Tg(n,m)*(m!/n!)*x^n, where gd(x) is Gudermannian function, Tg(n+1,m+1)=T(n,m). - _Vladimir Kruchinin_, Dec 18 2011
%C A147309 The Bell transform of abs(E(n)), E(n) the Euler numbers. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%H A147309 G. C. Greubel, <a href="/A147309/b147309.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A147309 From _Peter Bala_, Jan 26 2011: (Start)
%F A147309 GENERATING FUNCTION
%F A147309 The e.g.f., upon including a constant term of '1', is given by:
%F A147309 (1) F(x,t) = (tan(t) + sec(t))^x = Sum_{n>=0} Z(n,x)*t^n/n! = 1 + x*t + x^2*t^2/2! + (x+x^3)*t^3/3! + ....
%F A147309 Other forms include
%F A147309 (2) F(x,t) = exp(x*arcsinh(tan(t))) = exp(2*x*arctanh(tan(t/2))).
%F A147309 (3) F(x,t) = exp(x*(t + t^3/3! + 5*t^5/5! + 61*t^7/7! + ...)),
%F A147309 where the coefficients [1,1,5,61,...] are the secant or zig numbers A000364.
%F A147309 ROW GENERATING POLYNOMIALS
%F A147309 One easily checks from (1) that
%F A147309 d/dt(F(x,t)) = 1/2*x*(F(x-1,t) + F(x+1,t))
%F A147309 and so the row generating polynomials Z(n,x) satisfy the recurrence relation
%F A147309 (4) Z(n+1,x) = 1/2*x*{Z(n,x-1) + Z(n,x+1)}.
%F A147309 The e.g.f. for the odd-indexed row polynomials is
%F A147309 (5) sinh(x*arcsinh(tan(t))) = Sum_{n>=0} Z(2n+1,x)*t^(2n+1)/(2n+1)!.
%F A147309 The e.g.f. for the even-indexed row polynomials is
%F A147309 (6) cosh(x*arcsinh(tan(t))) = Sum_{n>=0} Z(2n,x)*t^(2n)/(2n)!.
%F A147309 From sinh(2*x) = 2*sinh(x)*cosh(x) we obtain the identity
%F A147309 (7) Z(2n+1,2*x) = 2*Sum_{k=0..n} binomial(2n+1,2k)*Z(2k,x)*Z(2n-2k+1,x).
%F A147309 The zeros of Z(n,x) lie on the imaginary axis (use (4) and adapt the proof given in A185417 for the zeros of the polynomial S(n,x)).
%F A147309 BINOMIAL EXPANSION
%F A147309 The form of the e.g.f. shows that {Z(n,x)} n >= 0 is a sequence of polynomials of binomial type. In particular, we have the expansion
%F A147309 (8) Z(n,x+y) = Sum_{k=0..n} binomial(n,k)*Z(k,x)*Z(n-k,y).
%F A147309 The delta operator D* associated with this binomial type sequence is
%F A147309 (9) D* = D - D^3/3! + 5*D^5/5! - 61*D^7/7! + 1385*D^9/9! - ..., and satisfies
%F A147309 the relation
%F A147309 (10) tan(D*)+sec(D*) = exp(D).
%F A147309 The delta operator D* acts as a lowering operator on the zigzag polynomials:
%F A147309 (11) (D*)Z(n,x) = n*Z(n-1,x).
%F A147309 ANALOG OF THE LITTLE FERMAT THEOREM
%F A147309 For integer x and odd prime p
%F A147309 (12) Z(p,x) = (-1)^((p-1)/2)*x (mod p).
%F A147309 More generally, for k = 1,2,3,...
%F A147309 (13) Z(p+k-1,x) = (-1)^((p-1)/2)*Z(k,x) (mod p).
%F A147309 RELATIONS WITH OTHER SEQUENCES
%F A147309 Row sums [1,1,2,5,16,61,...] are the zigzag numbers A000111(n) for n >= 1.
%F A147309 Column 1 (with 0's omitted) is the sequence of Euler numbers A000364.
%F A147309 A145876(n,k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1,k-j)*Z(n,j).
%F A147309 A147315(n-1,k-1) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*Z(n,j).
%F A147309 A185421(n,k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*Z(n,j).
%F A147309 A012123(n) = (-i)^n*Z(n,i) where i = sqrt(-1). A012259(n) = 2^n*Z(n,1/2).
%F A147309 (End)
%F A147309 T(n,m) = Sum(i=0..n-m, s(i)/(n-i)!*Sum(k=m..n-i, A147315(n-i,k)*Stirling1(k,m))), m>0, T(n,0) = s(n), s(n)=[1,0,1,0,5,0,61,0,1385,0,50521,...] (see A000364). - _Vladimir Kruchinin_, Mar 10 2011
%e A147309 Triangle begins
%e A147309    1;
%e A147309    0,  1;
%e A147309    1,  0,   1;
%e A147309    0,  4,   0,  1;
%e A147309    5,  0,  10,  0,  1;
%e A147309    0, 40,   0, 20,  0, 1;
%e A147309   61,  0, 175,  0, 35, 0, 1;
%p A147309 Z := proc(n, x) option remember;
%p A147309 description 'zigzag polynomials Z(n, x)'
%p A147309 if n = 0 return 1 else return 1/2*x*(Z(n-1, x-1)+Z(n-1, x+1)) end proc:
%p A147309 with(PolynomialTools):
%p A147309 for n from 1 to 10 CoefficientList(Z(n, x), x); end do; # _Peter Bala_, Jan 26 2011
%t A147309 t[n_, k_] := SeriesCoefficient[ 2^k*ArcTan[(E^x - 1)/(E^x + 1)]^k*n!/k!, {x, 0, n}]; Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten // Abs (* _Jean-François Alcover_, Jan 23 2015 *)
%o A147309 (PARI) T(n, k)=local(X); if(k<1 || k>n, 0, X=x+x*O(x^n); n!*polcoeff(polcoeff((tan(X)+1/cos(X))^y, n), k)) \\ _Paul D. Hanna_, Feb 06 2011
%o A147309 (Sage)
%o A147309 R = PolynomialRing(QQ, 'x')
%o A147309 @CachedFunction
%o A147309 def zzp(n, x) :
%o A147309     return 1 if n == 0 else x*(zzp(n-1, x-1)+zzp(n-1, x+1))/2
%o A147309 def A147309_row(n) :
%o A147309     x = R.gen()
%o A147309     L = list(R(zzp(n, x)))
%o A147309     del L[0]
%o A147309     return L
%o A147309 for n in (1..10) : print(A147309_row(n)) # _Peter Luschny_, Jul 22 2012
%o A147309 (Sage) # uses[bell_matrix from A264428]
%o A147309 # Alternative: Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A147309 bell_matrix(lambda n: abs(euler_number(n)), 10) # _Peter Luschny_, Jan 18 2016
%Y A147309 Cf. A000111 (row sums), A000364, A012123, A012259, A145876, A147315, A185415, A185417, A185419, A185421, A185424. - _Peter Bala_, Jan 26 2011
%K A147309 nonn,tabl
%O A147309 0,8
%A A147309 _Paul Barry_, Nov 05 2008