Real-rooted atlas

A243662 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(6 + 28\,x + 48\,x^{2} + 36\,x^{3} + 10\,x^{4} - 27\,n - 118\,n x - 194\,n x^{2} - 142\,n x^{3} - 39\,n x^{4} + 27\,n^{2} + 102\,n^{2} x + 140\,n^{2} x^{2} + 82\,n^{2} x^{3} + 17\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(3\,x + 14\,x^{2} + 24\,x^{3} + 18\,x^{4} + 5\,x^{5} - 9\,n x - 44\,n x^{2} - 78\,n x^{3} - 60\,n x^{4} - 17\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
312322165
455220290126691
52731456288625208187953
614289520252003311021595559496447
7775262016205428360672354008184192396931213761
8432634037881606374353115046323903626910156933028951015702715

All 15 rows

Technical details
Node id
n-5952c725ac8f7dc7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ba125e795100165f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       3,        4;
   3:      12,       32,       21;
   4:      55,      220,      290,       126;
   5:     273,     1456,     2886,      2520,       818;
   6:    1428,     9520,    25200,     33110,     21595,      5594;
   7:    7752,    62016,   205428,    360672,    354008,    184192,     39693;
   8:   43263,   403788,  1606374,   3531150,   4632390,   3626910,   1569330,    289510;
   9:  246675,  2631200, 12219900,  32274704,  53023740,  55489280,  36124200,  13376480,   2157150;
  10: 1430715, 17168580, 91171080, 281209500, 555216090, 727714020, 633200304, 352716408, 114139278, 16348960;
%F Formula
The row polynomials satisfy (4*n^2+2*n+(8*n^2+4*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (27*n^2-27*n+6+(102*n^2-118*n+28)*x+(140*n^2-194*n+48)*x^2+(82*n^2-142*n+36)*x^3+(17*n^2-39*n+10)*x^4)*P(n-1,x) - ((9*n-3)*x+(44*n-14)*x^2+(78*n-24)*x^3+(60*n-18)*x^4+(17*n-5)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 - 27*n + 6 + (102*n^2 - 118*n + 28)*x + (140*n^2 - 194*n + 48)*x^2 + (82*n^2 - 142*n + 36)*x^3 + (17*n^2 - 39*n + 10)*x^4)*P[n - 1, x] - ((9*n - 3)*x + (44*n - 14)*x^2 + (78*n - 24)*x^3 + (60*n - 18)*x^4 + (17*n - 5)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (8*n^2 + 4*n)*x + (4*n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((27*n^2 - 27*n + 6 + (102*n^2 - 118*n + 28)*x + (140*n^2 - 194*n + 48)*x^2 + (82*n^2 - 142*n + 36)*x^3 + (17*n^2 - 39*n + 10)*x^4)*v[i - 1] - ((9*n - 3)*x + (44*n - 14)*x^2 + (78*n - 24)*x^3 + (60*n - 18)*x^4 + (17*n - 5)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (8*n^2 + 4*n)*x + (4*n^2 + 2*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.