OFFSET
1,2
COMMENTS
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.
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 numerators 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. - Johannes W. Meijer, Oct 16 2009
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
FORMULA
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.
T(n,k) = (-1)^(n+k)*k*Stirling1(n+1,k+1). - Johannes W. Meijer, Oct 16 2009
E.g.f.: 1/(1 - z)^(x+1)*log(1/(1 - z)). Cf. A028421. - Peter Bala, Jan 06 2015
EXAMPLE
Polynomials begin:
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, ...
MAPLE
with(combinat): A074246 := proc(n, m): (-1)^(n+m)*binomial(m, 1)*stirling1(n+1, m+1) end: seq(seq(A074246(n, m), m=1..n), n=1..9); # Johannes W. Meijer, Oct 16 2009, Revised Sep 09 2012
MATHEMATICA
p[n_, x_] := Sum[1/(k+x), {k, 1, n}] Product[k+x, {k, 1, n}] ; Flatten[Table[ CoefficientList[ p[n, x] // Simplify[#, ComplexityFunction -> Length] &, x], {n, 1, 9}]] (* Jean-François Alcover, May 04 2011 *)
PROG
(PARI) P(n) = Vecrev(sum(k=1, n, prod(k=1, n, (k+x))/(k+x)));
for (n=1, 10, print(P(n))) \\ Michel Marcus, Jan 22 2017
CROSSREFS
KEYWORD
AUTHOR
Paul D. Hanna, Sep 19 2002
STATUS
approved