Real-rooted atlas

A008278 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
336110
442112138
556075201161
66155360195301747
77378150514004204213753
8888957968505420079856120253

All 15 rows

Technical details
Node id
n-87bd59d289a03b86 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9e6242213e8ef38e

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,    1;
   3:  3,    6,     1;
   4:  4,   21,    12,      1;
   5:  5,   60,    75,     20,      1;
   6:  6,  155,   360,    195,     30,      1;
   7:  7,  378,  1505,   1400,    420,     42,     1;
   8:  8,  889,  5796,   8505,   4200,    798,    56,    1;
   9:  9, 2040, 21175,  46620,  34755,  10584,  1386,   72,  1;
  10: 10, 4599, 74640, 238735, 255150, 114135, 23520, 2250, 90, 1;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^2-n+(n^2-n)*x+n*x^2)*P(n-2,x) - ((n^2-n)*x+n*x^2)*d/dx P(n-2,x) + (n^2-n+(2*n-1)*x)*P(n-1,x) + (n-1)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (n^2 - n)*x + n*x^2)*P[n - 2, x] - ((n^2 - n)*x + n*x^2)*D[P[n - 2, x], x] + (n^2 - n + (2*n - 1)*x)*P[n - 1, x] + (n - 1)*x*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] = 2 + x; for(n = 3, N, my(i = n); v[i] = (-(n^2 - n + (n^2 - n)*x + n*x^2)*v[i - 2] - ((n^2 - n)*x + n*x^2)*deriv(v[i - 2], x) + (n^2 - n + (2*n - 1)*x)*v[i - 1] + (n - 1)*x*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.