0,8

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,Artanh(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)).

W. Lang, First 10 rows.

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.

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).

Row sums: A003723. Unsigned row sums: A006229.

Sequence in context: A036852 A050327 A075120 this_sequence A111594 A105348 A016406

Adjacent sequences: A111590 A111591 A111592 this_sequence A111594 A111595 A111596

sign,easy,tabl

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 23 2005