A152650 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of certain polynomials P(n,x) given below.

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 9, 4, 1, 1, 2, 9, 8, 5, 1, 1, 4, 27, 32, 25, 6, 1, 1, 4, 81, 32, 125, 18, 7, 1, 1, 8, 81, 128, 625, 36, 49, 8, 1, 1, 2, 243, 256, 625, 54, 343, 32, 9, 1, 1, 4, 729, 1024, 3125, 324, 2401, 256, 81, 10, 1

Offset

0,5

Comments

Let the polynomials P be defined by P(0,x)=u(0), P(n,x)= u(n) + x*Sum_{i=0..n-1} u(i)*P(n-i-1,x) and coefficients u(i)=1/i!. These u are reminiscent of the Taylor expansion of exp(x). Then P(n,x) = Sum_{k=0..n} c(n,k)*x^k.

n!*P(n,x) are the row polynomials of A152818. - Peter Bala, Oct 09 2011

Conjecture: All roots of P(n,x) are real, hence negative. - Jean-François Alcover, Oct 10 2012

Links

Table of n, a(n) for n=0..65.

Jean-François Alcover, Plot showing roots of P(20,x)

Example

The triangle c(n,k) and polynomials start in row n = 0 as:
1 = 1;
1, 1 = 1 + x;
1/2, 2, 1 = 1/2 + 2*x + x^2;
1/6, 2, 3, 1, = 1/6+2*x+3*x^2+x^3
1/24, 4/3, 9/2, 4, 1, = 1/24 + 4/3*x + 9/2*x^2 + 4*x^3 + x^4;
1/120, 2/3, 9/2, 8, 5, 1, = 1/120 + 2/3*x + 9/2*x^2 + 8*x^3 + 5*x^4 + x^5;
1/720, 4/15, 27/8, 32/3, 25/2, 6, 1, = 1/720 + 4/15*x + 27/8*x^2 + 32/3*x^3 + 25/2*x^4 + 6*x^5 + x^6;
1/5040, 4/45, 81/40, 32/3, 125/6, 18, 7, 1 = 1/5040 + 4/45*x + 81/40*x^2 + 32/3*x^3 + 125/6*x^4 + 18*x^5 + 7*x^6 + x^7;

Maple

u := proc(i) 1/i! end:
P := proc(n, x) option remember ; if n =0 then u(0); else u(n)+x*add( u(i)*procname(n-1-i, x), i=0..n-1) ; expand(%) ; fi; end:
A152650 := proc(n, k) p := P(n, x) ; numer(coeftayl(p, x=0, k)) ; end:
seq(seq(A152650(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Aug 24 2009

Mathematica

ClearAll[u, p]; u[n_] := 1/n!; p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n-i-1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Numerator (* Jean-François Alcover, Oct 02 2012 *)

Crossrefs

Cf. A152656 (denominators), A140749, A141412, A141904, A142048. A152818.

Sequence in context: A145972 A215204 A260876 * A184219 A180262 A161789

Adjacent sequences: A152647 A152648 A152649 * A152651 A152652 A152653

Keyword

nonn,frac,tabl

Author

Paul Curtz, Dec 10 2008

Extensions

Edited and extended by R. J. Mathar, Aug 24 2009

Status

approved