Real-rooted atlas

A120986 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-8\,n - 22\,n x - 20\,n x^{2} - 6\,n x^{3} + 14\,n^{2} + 46\,n^{2} x + 50\,n^{2} x^{2} + 18\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 6\,x - 12\,x^{2} - 10\,x^{3} - 3\,x^{4} + n + 12\,n x + 30\,n x^{2} + 28\,n x^{3} + 9\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
35161233
4147011055249
5422887027282732033
61321155385061604760142817485
74294576194484243250388310087752156033
814301801893366259350418950395010201894432631431281

All 15 rows

Technical details
Node id
n-8f2b5ae45609013a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-031025df87457cd9

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:     2,      3;
   3:     5,     16,      12;
   4:    14,     70,     110,      55;
   5:    42,    288,     702,     728,      273;
   6:   132,   1155,    3850,    6160,     4760,     1428;
   7:   429,   4576,   19448,   42432,    50388,    31008,     7752;
   8:  1430,  18018,   93366,  259350,   418950,   395010,   201894,    43263;
   9:  4862,  70720,  433160, 1466080,  3010700,  3853696,  3010700,  1315600,  246675;
  10: 16796, 277134, 1961256, 7845024, 19612560, 31870410, 33745140, 22496760, 8584290, 1430715;
%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) = (14*n^2-8*n+(46*n^2-22*n)*x+(50*n^2-20*n)*x^2+(18*n^2-6*n)*x^3)*P(n-1,x) + (n-1+(12*n-6)*x+(30*n-12)*x^2+(28*n-10)*x^3+(9*n-3)*x^4)*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[((14*n^2 - 8*n + (46*n^2 - 22*n)*x + (50*n^2 - 20*n)*x^2 + (18*n^2 - 6*n)*x^3)*P[n - 1, x] + (n - 1 + (12*n - 6)*x + (30*n - 12)*x^2 + (28*n - 10)*x^3 + (9*n - 3)*x^4)*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] = ((14*n^2 - 8*n + (46*n^2 - 22*n)*x + (50*n^2 - 20*n)*x^2 + (18*n^2 - 6*n)*x^3)*v[i - 1] + (n - 1 + (12*n - 6)*x + (30*n - 12)*x^2 + (28*n - 10)*x^3 + (9*n - 3)*x^4)*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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.