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

Author

Paul D. Hanna, Sep 19 2002

Keywords

• easy
• nice
• nonn
• tabl

Comments

The n-th row polynomial, P(n,x), has ordered zeros {z_k < z_(k+1), 0

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

Examples

			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, ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

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

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

Programs

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 *)

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

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.