Real-rooted atlas

A008277 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
315511
419221547
5114619952227
61201353854712031215
71272601140241623868777107
8135455283591561547012867414044959

All 15 rows

Technical details
Node id
n-72fe1904e30238b0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-1090d35600282d8c

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: 1,  2;
   3: 1,  5,    5;
   4: 1,  9,   22,    15;
   5: 1, 14,   61,    99,    52;
   6: 1, 20,  135,   385,   471,    203;
   7: 1, 27,  260,  1140,  2416,   2386,    877;
   8: 1, 35,  455,  2835,  9156,  15470,  12867,   4140;
   9: 1, 44,  742,  6230, 28441,  72590, 102215,  73681,  21147;
  10: 1, 54, 1146, 12474, 76503, 274386, 578929, 700311, 446620, 115975;
%F Formula
The row polynomials satisfy P(n,x) = (1+n*x+(n-2)*x^2)*P(n-1,x) - (x^2+x^3)*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[(1 + n*x + (n - 2)*x^2)*P[n - 1, x] - (x^2 + x^3)*D[P[n - 1, x], x]];
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] = (1 + n*x + (n - 2)*x^2)*v[i - 1] - (x^2 + x^3)*deriv(v[i - 1], x)); 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

Candidate equalities ?

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.