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

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, 0, 560, 0, 56, 0, 1, 1385, 0, 4996, 0, 1470, 0, 84, 0, 1, 0, 24320, 0, 22720, 0, 3360, 0, 120, 0, 1

Offset

0,8

Comments

Production array is [cosh(x),x] beheaded. Inverse is A147308. Row sums are A000111(n+1).

Unsigned version of A147308. - N. J. A. Sloane, Nov 07 2008

From Peter Bala, Jan 26 2011: (Start)

Define a polynomial sequence {Z(n,x)} n >= 0 by means of the recursion

(1)... Z(n+1,x) = 1/2*x*{Z(n,x-1)+Z(n,x+1)}

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

... Z(1,x) = x

... Z(2,x) = x^2

... Z(3,x) = x + x^3

... Z(4,x) = 4*x^2 + x^4

... Z(5,x) = 5*x + 10*x^3 + x^5.

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.

{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:

(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],

where {B_k} k >= 0 = [1, -1/2, 1/6, 0, -1/30, ...] is the sequence of Bernoulli numbers.

For similarly defined polynomial sequences to Z(n,x) see A185415, A185417 and A185419. See also A185424.

(End)

[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

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

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

From Peter Bala, Jan 26 2011: (Start)

GENERATING FUNCTION

The e.g.f., upon including a constant term of '1', is given by:

(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! + ....

Other forms include

(2) F(x,t) = exp(x*arcsinh(tan(t))) = exp(2*x*arctanh(tan(t/2))).

(3) F(x,t) = exp(x*(t + t^3/3! + 5*t^5/5! + 61*t^7/7! + ...)),

where the coefficients [1,1,5,61,...] are the secant or zig numbers A000364.

ROW GENERATING POLYNOMIALS

One easily checks from (1) that

d/dt(F(x,t)) = 1/2*x*(F(x-1,t) + F(x+1,t))

and so the row generating polynomials Z(n,x) satisfy the recurrence relation

(4) Z(n+1,x) = 1/2*x*{Z(n,x-1) + Z(n,x+1)}.

The e.g.f. for the odd-indexed row polynomials is

(5) sinh(x*arcsinh(tan(t))) = Sum_{n>=0} Z(2n+1,x)*t^(2n+1)/(2n+1)!.

The e.g.f. for the even-indexed row polynomials is

(6) cosh(x*arcsinh(tan(t))) = Sum_{n>=0} Z(2n,x)*t^(2n)/(2n)!.

From sinh(2*x) = 2*sinh(x)*cosh(x) we obtain the identity

(7) Z(2n+1,2*x) = 2*Sum_{k=0..n} binomial(2n+1,2k)*Z(2k,x)*Z(2n-2k+1,x).

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)).

BINOMIAL EXPANSION

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

(8) Z(n,x+y) = Sum_{k=0..n} binomial(n,k)*Z(k,x)*Z(n-k,y).

The delta operator D* associated with this binomial type sequence is

(9) D* = D - D^3/3! + 5*D^5/5! - 61*D^7/7! + 1385*D^9/9! - ..., and satisfies

the relation

(10) tan(D*)+sec(D*) = exp(D).

The delta operator D* acts as a lowering operator on the zigzag polynomials:

(11) (D*)Z(n,x) = n*Z(n-1,x).

ANALOG OF THE LITTLE FERMAT THEOREM

For integer x and odd prime p

(12) Z(p,x) = (-1)^((p-1)/2)*x (mod p).

More generally, for k = 1,2,3,...

(13) Z(p+k-1,x) = (-1)^((p-1)/2)*Z(k,x) (mod p).

RELATIONS WITH OTHER SEQUENCES

Row sums [1,1,2,5,16,61,...] are the zigzag numbers A000111(n) for n >= 1.

Column 1 (with 0's omitted) is the sequence of Euler numbers A000364.

A145876(n,k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1,k-j)*Z(n,j).

A147315(n-1,k-1) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*Z(n,j).

A185421(n,k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*Z(n,j).

A012123(n) = (-i)^n*Z(n,i) where i = sqrt(-1). A012259(n) = 2^n*Z(n,1/2).

(End)

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

Example

Triangle begins
   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;

Maple

Z := proc(n, x) option remember;
description 'zigzag polynomials Z(n, x)'
if n = 0 return 1 else return 1/2*x*(Z(n-1, x-1)+Z(n-1, x+1)) end proc:
with(PolynomialTools):
for n from 1 to 10 CoefficientList(Z(n, x), x); end do; # Peter Bala, Jan 26 2011

Mathematica

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 *)

Prog

(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
(Sage)
R = PolynomialRing(QQ, 'x')
@CachedFunction
def zzp(n, x) :
    return 1 if n == 0 else x*(zzp(n-1, x-1)+zzp(n-1, x+1))/2
def A147309_row(n) :
    x = R.gen()
    L = list(R(zzp(n, x)))
    del L[0]
    return L
for n in (1..10) : print(A147309_row(n)) # Peter Luschny, Jul 22 2012
(Sage) # uses[bell_matrix from A264428]
# Alternative: Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: abs(euler_number(n)), 10) # Peter Luschny, Jan 18 2016

Crossrefs

Cf. A000111 (row sums), A000364, A012123, A012259, A145876, A147315, A185415, A185417, A185419, A185421, A185424. - Peter Bala, Jan 26 2011

Sequence in context: A186759 A065623 A178103 * A147308 A110064 A021253

Adjacent sequences: A147306 A147307 A147308 * A147310 A147311 A147312

Keyword

nonn,tabl

Author

Paul Barry, Nov 05 2008

Status

approved