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A111594 Triangle of arctanh numbers. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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.

W. 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).

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-Fran├žois Alcover, Jun 22 2018, after Peter Luschny *)

PROG

(Sage)

# The function riordan_array is defined in 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: A050327 A075120 A111593 * A322549 A237996 A203951

Adjacent sequences:  A111591 A111592 A111593 * A111595 A111596 A111597

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Aug 23 2005

STATUS

approved

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Last modified March 25 22:28 EDT 2019. Contains 321477 sequences. (Running on oeis4.)