OFFSET
0,8
COMMENTS
Sheffer triangle associated to Sheffer triangle A060081.
For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
In the umbral calculus (see the S. Roman reference) this triangle would be called associated for (1,arctanh(y)).
Without the n=0 row and m=0 column and unsigned, this is the Jabotinsky triangle A059419.
The inverse matrix of A with elements a(n,m), n,m>=0, is A111594.
The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060081, 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.
The row polynomials p(n,x) (defined above) have e.g.f. exp(x*tanh(y)).
Exponential Riordan array [1, tanh(x)], inverse of [1, arctanh(x)] which is A111594. - Paul Barry, May 30 2010
Also the Bell transform of A155585(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
LINKS
FORMULA
E.g.f. for column m>=0: ((tanh(x))^m)/m!.
a(n, m) = coefficient of x^n of ((tanh(x))^m)/m!, n>=m>=0, else 0.
a(n, m) = a(n-1, m-1) - (m+1)*m*a(n-1, m+1), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n<m.
T(n,m) = Sum_{k=0..n-m} binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(n-k-m)*stirling2(n,k+m))/m!, T(0,0)=1. - Vladimir Kruchinin, Jun 09 2011
With e.g.f. exp(x*tanh(t)) = sum(n>= 0, P(n,x)*t^n/n!), the lowering operator is L = arctanh(d/dx) = d/dx + (1/3)(d/dx)^3 + (1/5)(d/dx)^5 + ..., and the raising operator is R = x [1 - (d/dx)^2], where L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x), since the sequence is a binomial Sheffer sequence. - Tom Copeland, Oct 01 2015
The raising operator R = x - x D^2 in matrix form acting on an o.g.f. (formal power series) is the transpose of the production matrix M below. The linear term x is the diagonal of ones after transposition. The other transposed diagonal (A002378) comes from -x D^2 x^n = -n * (n-1) x^(n-1). Then P(n,x) = (1,x,x^2,..) M^n (1,0,0,..)^T. - Tom Copeland, Aug 17 2016
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 A060081:
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).
From Paul Barry, May 30 2010: (Start)
Triangle begins
1;
0, 1;
0, 0, 1;
0, -2, 0, 1;
0, 0, -8, 0, 1;
0, 16, 0, -20, 0, 1;
0, 0, 136, 0, -40, 0, 1;
0, -272, 0, 616, 0, -70, 0, 1;
0, 0, -3968, 0, 2016, 0, -112, 0, 1;
Production matrix begins
0, 1;
0, 0, 1;
0, -2, 0, 1;
0, 0, -6, 0, 1;
0, 0, 0, -12, 0, 1;
0, 0, 0, 0, -20, 0, 1;
0, 0, 0, 0, 0, -30, 0, 1;
0, 0, 0, 0, 0, 0, -42, 0, 1;
0, 0, 0, 0, 0, 0, 0, -56, 0, 1; (End)
MAPLE
# The function BellMatrix is defined in A264428.
BellMatrix(n -> 2^(n+1)*euler(n+1, 1), 9); # Peter Luschny, Jan 26 2016
MATHEMATICA
t[0, 0] = 1; t[n_, m_] := Sum[ Binomial[k+m-1, m-1]*(k+m)!*(-1)^(k)*2^(n-k-m)*StirlingS2[n, k+m], {k, 0, n-m}]/m!; Table[t[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[2^(#+1)*EulerE[#+1, 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
PROG
(Maxima)
T(n, m):=if n=0 and m=0 then 1 else sum(binomial(k+m-1, m-1)*(k+m)!*(-1)^(k)*2^(n-k-m)*stirling2(n, k+m), k, 0, n-m)/m!; /* Vladimir Kruchinin, Jun 09 2011 */
(Sage) # uses[riordan_array from A256893]
riordan_array(1, tanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Aug 23 2005
STATUS
approved