1,2

The n-th row polynomial, P(n,x), has ordered zeros {z_k < z_(k+1), 0<k<n} that satisfy z_k + z_(n-k) = -(n+1) and integerpart(z_k) = -k. For even rows, polynomial P(2n,x) has zero z_n = -(n+1)/2. Example: at n=6, P(6,x) has zeros z_1 = -1.336553473264694, z_2 = -2.426299641757407, z_3 = -3.5, z_4 = -4.573700358242594, z_5 = -5.663446526735307.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

The higher order exponential integrals E(x,m,n) are defined in A163931 and the asymptotic expansion of E(x,m=2,n) can be found in A028421. We determined with the latter that E(x,m=2,n+1) = (exp(-x)/x^2)*(1 - (3+2*n)/x + (11+12*n+3*n^2)/x^2 - (50+70*n+30*n^2+ 4*n^3)/x^3 + .... ). The polynomial coefficients in the nominators lead to the coefficients of the triangle given above. The numerators of the o.g.f.s. of the right hand columns of this triangle lead for z = 1 to A001147.

(End)

First column is A000254 (Stirling numbers of first kind s(n, 2): a(n+1)=(n+1)*a(n)+n!), while sum of rows is A001705 (generalized Stirling numbers). Also related to Harmonic numbers: P(n, 0)=n!*H(n), H(n)=harmonic number.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

a(n,m) = (-1)^(n+m)*binomial(m,1)*stirling1(n+1,m+1)

(End)

P(1,x)=1, P(2,x)=3 + 2x, P(3,x)=11 + 12x + 3x^2, P(4,x)=50 + 70x + 30x^2 + 4x^3, P(5,x)=274 + 450x + 255x^2 + 60x^3 + 5x^4, P(6,x)=1764 + 3248x + 2205x^2 + 700x^3 + 105x^4 + 6x^5, P(7,x)=13068 + 26264x + 20307x^2 + 7840x^3 + 1610x^4 + 168x^5 + 7x^6, P(8,x)=109584 + 236248x + 201852x^2 + 89796x^3 + 22680x^4 + 3276x^5 + 252x^6 + 8x^7, P(9,x)=1026576 + 2345400x + 2171040x^2 + 1077300x^3 + 316365x^4 + 56700x^5 + 6090x^6 + 360x^7 + 9x^8, P(10,x)=10628640 + 25507152x + 25228500x^2 + 13667720x^3 + 4510275x^4 + 946638x^5 + 127050x^6 + 10560x^7 + 495x^8 + 10x^9.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

nmax:=9; mmax:=nmax: with(combinat, stirling1): for n from 1 to nmax do for m from 1 to n do a(n, m):=(-1)^(n+m)*binomial(m, 1)*stirling1(n+1, m+1) od: od: T:=1: for n from 1 to nmax do for m from 1 to n do a(T):=a(n, m); T:=T+1: od: od: seq(a(n), n=1..T-1);

(End)

See references and formulas at A000254, A001705.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

A027480 is the second right hand column.

(End)

Sequence in context: A072634 A086194 A159610 this_sequence A134426 A122672 A052973

Adjacent sequences: A074243 A074244 A074245 this_sequence A074247 A074248 A074249

easy,nice,nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 19 2002