1,3

The row polynomials p(n,x) := sum(a(n,m)x^m,m=0..n-1), n>=1, are obtained from ((Eu(x)^n)*(x-1)^n)/(n*x), where Eu(x) := xd/dx is the Euler-derivative with respect to x.

The row polynomials p(n,y) := sum(a(n,m)y^m,m=0..n-1), n>=1, are also obtained from diff(((exp(x)-1)^m)/m,x$m)/exp(x) after replacement of exp(x) by y. Here diff(f(x),x$m), m>=1, denotes m-fold differentiation of f(x) with respect to x.

b(k,m,n) := sum((a(m,p)*((p+1)*k)^n)/(m-1)!,p=0..m-1), n>=0, has g.f. 1/product(1-k*p*x,p=1..m) for k=1,2,... and m=1,2,...

The (signed) row sums give A000142(n-1), n>=1, (factorials) and (unsigned) A074932(n).

The (unsigned) columns give A000012 (powers of 1), 2*A001787(n+1), (3^2)*A027472(n), (4^3)*A038846(n-1), (5^4)A036071(n-5), (6^5)*A036084(n-6), (7^6)* A036226(n-7), (8^7)*A053107(n-8) for m=0..7.

a(n, m)= ((-1)^(n-m-1)) binomial(n-1, m)*(m+1)^(n-1), n>=m+1>=1, else 0.

G.f. for m-th column: ((m+1)^m)(x/(1+(m+1)*x))^(m+1), m>=0.

E.g.f.: -LambertW(-x*y*exp(-x))/(1+LambertW(-x*y*exp(-x)))/x. - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 13 2008

[1];[ -1,2];[1,-8,9];[ -1,24,-81,64]; ...; p(2,x)=-1+2x=(1/(2*x))*x*(d/dx)*x*(d/dx)*(x-1)^2.

Cf. A075510-2, A074932, A075515-16, A075906-25, A076002-13.

Sequence in context: A086657 A036296 A078105 this_sequence A011019 A164662 A007026

Adjacent sequences: A075510 A075511 A075512 this_sequence A075514 A075515 A075516

sign,tabl,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 02, 2002