A074246 Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.

1, 3, 2, 11, 12, 3, 50, 70, 30, 4, 274, 450, 255, 60, 5, 1764, 3248, 2205, 700, 105, 6, 13068, 26264, 20307, 7840, 1610, 168, 7, 109584, 236248, 201852, 89796, 22680, 3276, 252, 8, 1026576, 2345400, 2171040, 1077300, 316365, 56700, 6090, 360, 9

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

See references and formulas at A000254, A001705. Cf. A028421.

A027480 is the second right hand column. - Johannes W. Meijer, Oct 16 2009

Sequence in context: A086194 A258386 A159610 * A134426 A122672 A194608

Adjacent sequences: A074243 A074244 A074245 * A074247 A074248 A074249

Keyword

easy,nice,nonn,tabl

Author

Paul D. Hanna, Sep 19 2002

Status

approved