Real-rooted atlas

A008278 → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(n - 2\,n x^{2} + n^{2} + 4\,n^{2} x + 3\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-n x - 2\,n x^{2} - n x^{3} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 2\,x - 2\,n - 4\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(1 + 2\,x + x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
31012325
43866334141
5161370285805901
674721502265104018566393
737531303417535113403500420749589
8202538264213589111424051100112989458416377

All 15 rows

Technical details
Node id
n-1cc0cc02ac61dbcf · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a464fc6e981d0b75

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,       2;
   3:     10,      12,       3;
   4:     38,      66,      33,        4;
   5:    161,     370,     285,       80,       5;
   6:    747,    2150,    2265,     1040,     185,       6;
   7:   3753,   13034,   17535,    11340,    3500,     420,      7;
   8:  20253,   82642,  135891,   114240,   51100,   11298,    945,      8;
   9: 116642,  548118, 1068585,  1111208,  657510,  217014,  35707,   2112,    9;
  10: 713130, 3799212, 8586717, 10677044, 7906500, 3513804, 890827, 111792, 4689, 10;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3-n^2-n+(2*n^3-4*n^2)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (n^2+(3*n^2-n)*x+(3*n^2-2*n)*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n^2-2*n+1+(2*n^2-4*n+2)*x)*P(n-1,x) - (n-1+(2*n-2)*x+(n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - n + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] + (n^2 + (3*n^2 - n)*x + (3*n^2 - 2*n)*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] - (n - 1 + (2*n - 2)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(n^3 - n^2 - n + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (n^2 + (3*n^2 - n)*x + (3*n^2 - 2*n)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x)*v[i - 1] - (n - 1 + (2*n - 2)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.