0,5

Without row n=0 and column m=0 this is, up to signs, the Lah triangle A008297.

The unsigned column sequences are (with leading zeros): A000142, A001286, A001754, A001755, A001777, A001778, A111597-A111600 for m=1..10.

The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A111595 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.

Exponential Riordan array [1,x/(1+x)]. Inverse of the exponential Riordan array [1,x/(1-x)], which is the unsigned version of A111596. - Paul Barry (pbarry(AT)wit.ie), Apr 12 2007

For the unsigned subtriangle without column nr. m=0 and row nr. n=0 see A105278.

Unsigned triangle also matrix product |S1|*S2 of Stirling number matrices.

The unsigned row polynomials are Lag(n,-x,-1), the associated Laguerre polynomials of order -1 with negated argument. See Gradshteyn and Ryzhik, Abramowitz and Stegun and Rota (Finite Operator Calculus) for extensive formulae. - Tom Copeland (tcjpn(AT)msn.com), Nov 17 2007, Sep 09 2008

An infinitesimal matrix generator for unsigned A111596 is given by A132792. - Tom Copeland (tcjpn(AT)msn.com), Nov 22 2007

From the formalism of A132792 and A133314 for n > k, unsigned A111596(n,k) = a(k) * a(k+1)...a(n-1) / (n-k)! = a generalized factorial, where a(n) = A002378(n) = n-th term of first subdiagonal of unsigned A111596. Hence Deutsch's remark in A002378 provides an interpretation of A111596(n,k) in terms of combinations of certain circular binary words. - Tom Copeland (tcjpn(AT)msn.com), Nov 22 2007

Given T(n,k)= A111596(n,k) and matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. An e.g.f. for the row polynomials of A is exp[ -a(.)*x^2*D_x] exp(x*t) = exp(x*t) exp[(.)!*Lag(.,-x*t,-1)*a(.)*(-x)], umbrally, where [(.)! Lag(.,x,-1)]^n = n! Lag(n,x,-1) is a normalized Laguerre polynomial of order -1. (Caution: sometimes the factor n! is included in the definition of the Laguerre polynomial (Rota convention), as in an earlier comment in this entry.) [From Tom Copeland (tcjpn(AT)msn.com), Aug 27 2008]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

W. Lang, First 10 rows.

E.g.f. m-th column: ((x/(1+x))^m)/m!, m>=0.

E.g.f. for row polynomials p(n, x) is exp(x*y/(1+y)).

a(n, m)= ((-1)^(n-m))*|A008297(n, m)| = ((-1)^(n-m))*(n!/m!)*binomial(n-1, m-1), n>=m>=1; a(0, 0)=1; else 0.

a(n, m) = -(n-1+m)*a(n-1, m) + a(n-1, m-1), n>=m>=0, a(n, -1):=0, a(0, 0)=1; a(n, m)=0 if n<m.

|a(n,m)|=sum(|S1(n,k)|*S2(k,m),k=m..n), n>=0. S2(n,m):=A048993. S1(n,m):=A048994. W. Lang, May 04, 2007.

Binomial convolution of row polynomials: p(3,x) = 6*x-6*x^2+x^3; p(2,x) = -2*x+x^2, p(1,x) = x, p(0,x) = 1,

together with those from A111595: s(3,x) = 9*x-6*x^2+x^3; s(2,x) = 1-2*x+x^2, s(1,x) = x, s(0,x) = 1; therefore

9*(x+y)-6*(x+y)^2+(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) = (6*y-6*y^2+y^3) + 3*x*(-2*y+y^2)

+ 3*(1-2*x+x^2)*y + 9*x-6*x^2+x^3.

Row sums: A111884. Unsigned row sums: A000262.

Adjacent sequences: A111593 A111594 A111595 this_sequence A111597 A111598 A111599

Sequence in context: A047922 A021830 A111184 this_sequence A129062 A163936 A117651

sign,easy,tabl

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