0,8

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*Artanh(y)).

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

W. Lang, First 10 rows.

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

a(n, m)= coefficient of x^n of ((Artanh(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.

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).

Row sums: A000246.

Sequence in context: A050327 A075120 A111593 this_sequence A105348 A016406 A129182

Adjacent sequences: A111591 A111592 A111593 this_sequence A111595 A111596 A111597

nonn,easy,tabl

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