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

Author

Paul Curtz, Dec 10 2008

Keywords

• nonn
• frac
• tabl

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

Examples

			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;
		

Links

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

Crossrefs

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

Programs

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 *)

Extensions

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