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%I #21 Mar 17 2019 03:46:47
%S 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,
%T 125,18,7,1,1,8,81,128,625,36,49,8,1,1,2,243,256,625,54,343,32,9,1,1,
%U 4,729,1024,3125,324,2401,256,81,10,1
%N Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of certain polynomials P(n,x) given below.
%C 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.
%C n!*P(n,x) are the row polynomials of A152818. - _Peter Bala_, Oct 09 2011
%C Conjecture: All roots of P(n,x) are real, hence negative. - _Jean-François Alcover_, Oct 10 2012
%H Jean-François Alcover, <a href="/A152650/a152650.pdf">Plot showing roots of P(20,x)</a>
%e The triangle c(n,k) and polynomials start in row n = 0 as:
%e 1 = 1;
%e 1, 1 = 1 + x;
%e 1/2, 2, 1 = 1/2 + 2*x + x^2;
%e 1/6, 2, 3, 1, = 1/6+2*x+3*x^2+x^3
%e 1/24, 4/3, 9/2, 4, 1, = 1/24 + 4/3*x + 9/2*x^2 + 4*x^3 + x^4;
%e 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;
%e 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;
%e 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;
%p u := proc(i) 1/i! end:
%p 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:
%p A152650 := proc(n,k) p := P(n,x) ; numer(coeftayl(p,x=0,k)) ; end:
%p seq(seq(A152650(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Aug 24 2009
%t 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 *)
%Y Cf. A152656 (denominators), A140749, A141412, A141904, A142048. A152818.
%K nonn,frac,tabl
%O 0,5
%A _Paul Curtz_, Dec 10 2008
%E Edited and extended by _R. J. Mathar_, Aug 24 2009
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