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A152650 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of certain polynomials P(n,x) given below. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified April 20 08:39 EDT 2019. Contains 322306 sequences. (Running on oeis4.)