A111594 Triangle of arctanh numbers.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 8, 0, 1, 0, 24, 0, 20, 0, 1, 0, 0, 184, 0, 40, 0, 1, 0, 720, 0, 784, 0, 70, 0, 1, 0, 0, 8448, 0, 2464, 0, 112, 0, 1, 0, 40320, 0, 52352, 0, 6384, 0, 168, 0, 1, 0, 0, 648576, 0, 229760, 0, 14448, 0, 240, 0, 1

Offset

0,8

Comments

Sheffer triangle associated to Sheffer triangle A060524.

For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.

The inverse matrix of A with elements a(n,m), n,m>=0, is given in A111593.

In the umbral calculus notation (see the S. Roman reference) this triangle would be called associated to (1,tanh(y)).

The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060524 satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.

Without the n=0 row and m=0 column and signed, this will become the Jabotinsky triangle A049218 (arctan numbers). For Jabotinsky matrices see the Knuth reference under A039692.

The row polynomials p(n,x) (defined above) have e.g.f. exp(x*arctanh(y)).

Exponential Riordan array [1, arctanh(x)] = [1, log(sqrt((1+x)/(1-x)))]. - Paul Barry, Apr 17 2008

Also the Bell transform of A005359. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

Links

Table of n, a(n) for n=0..65.

Wolfdieter Lang, First 10 rows.

Formula

E.g.f. for column m>=0: ((arctanh(x))^m)/m!.

a(n, m) = coefficient of x^n of ((arctanh(x))^m)/m!, n>=m>=0, else 0.

a(n, m) = a(n-1, m-1) + (n-2)*(n-1)*a(n-2, m), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n<m.

Example

Binomial convolution of row polynomials:
p(3,x)= 2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1,
together with those from A060524:
s(3,x)= 5*x+x^3; s(2,x)= 1+x^2, s(1,x)= x, s(0,x)= 1; therefore:
5*(x+y)+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = 2*y+y^3 + 3*x*y^2 + 3*(1+x^2)*y + (5*x+x^3).
Triangle begins:
  1;
  0,   1;
  0,   0,    1;
  0,   2,    0,   1;
  0,   0,    8,   0,    1;
  0,  24,    0,  20,    0,  1;
  0,   0,  184,   0,   40,  0,   1;
  0, 720,    0, 784,    0, 70,   0, 1;
  0,   0, 8448,   0, 2464,  0, 112, 0, 1;
...

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::even, n!, 0), 10); # Peter Luschny, Jan 27 2016

Mathematica

rows = 10;
t = Table[If[EvenQ[n], n!, 0], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

Prog

(SageMath) # uses[riordan_array from A256893]
riordan_array(1, atanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015

Crossrefs

Row sums: A000246.

Cf. A005359, A049218, A060524, A111593.

Sequence in context: A075120 A327751 A111593 * A322549 A349645 A237996

Adjacent sequences: A111591 A111592 A111593 * A111595 A111596 A111597

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Aug 23 2005

Status

approved