A147308 Riordan array [sech(x), arcsin(tanh(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, -50521, 0, 214445, 0

Offset

0,8

Comments

Production array is [cos(x),x] beheaded. Inverse is A147309. Row sums are A012123(n+1).

If signs are ignored this is identical to A147309. - N. J. A. Sloane, Nov 07 2008

The Bell transform of the Euler numbers. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..229

Formula

From Vladimir Kruchinin, Dec 18 2011: (Start)

[gd(x)]^m = sum(n>=m T(n,m)*m!/n!*x^n), where gd(x) is the Gudermannian function.

T(n,m) = sum(j=0..(n-m)/2, (sum(i=0..2*j, (2^(i)*Stirling1(i+m,m) *C(2*j+m-1,i+m-1))/(i+m)!)) *sum(k=0..n-2*j-m, (-1)^(k+j) *C(k+2*j+m-1,2*j+m-1) *(k+2*j+m)! *2^(-k-2*j) *Stirling2(n,k+2*j+m))), n>=m>=1.

(End)

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; if n = 0 then return 1: else return 1/2*x*(Z(n-1, x-1)+Z(n-1, x+1)): fi:end:
with(PolynomialTools): for n from 1 to 10 do for k from 1 to n do printf("%d, ", (-1)^floor((n-k)/2)*coeff(Z(n, x), x, k)):od: printf("\n"):od: # Nathaniel Johnston, Apr 21 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, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 23 2015 *)
Z[n_, x_] := Z[n, x] = If[n == 0, 1, x*(Z[n-1, x-1] + Z[n-1, x+1])/2 // Simplify]; t[n_, k_] := (-1)^Floor[(n-k)/2]*Coefficient[Z[n, x], x, k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 27 2015, after Maple *)

Prog

(Sage) # uses[bell_matrix from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(euler_number, 10) # Peter Luschny, Jan 18 2016

Crossrefs

Sequence in context: A065623 A178103 A147309 * A110064 A021253 A136586

Adjacent sequences: A147305 A147306 A147307 * A147309 A147310 A147311

Keyword

easy,sign,tabl

Author

Paul Barry, Nov 05 2008

Status

approved