Real-rooted atlas

A253283 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(4 - n\right)\,P_{n-3}(x) + \left(-23 - 14\,x + 7\,n + 4\,n x\right)\,P_{n-2}(x) + \left(-14\,x + 4\,n x\right)\,P_{n-2}'(x) + \left(12 + 6\,x - 7\,n - 4\,n x\right)\,P_{n-1}(x) + \left(6\,x - 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)
\(P_{3}(x) = 5 + 11\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
3511319
4251097210216
512910391265410352878
668197611966011200217012643598
73653913072874272567608484010962462735411
81982585436540628285353026268611057981053592171613611272

All 15 rows

Technical details
Node id
n-d58a953b91eff3a7 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-be3c7f562b31849c

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,        1;
   3:      5,       11,         3;
   4:     25,      109,        72,         10;
   5:    129,     1039,      1265,        410,         35;
   6:    681,     9761,     19660,      11200,       2170,       126;
   7:   3653,    91307,    287427,     256760,      84840,     10962,       462;
   8:  19825,   854365,   4062828,    5353026,    2686110,    579810,     53592,     1716;
   9: 108545,  8013215,  56318817,  105397194,   75427275,  24201870,   3681678,   255684,    6435;
  10: 598417, 75396673, 771767928, 2000574400, 1963569300, 878906028, 195723528, 22119240, 1196910, 24310;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = -(n-4)*P(n-3,x) + (7*n-23+(4*n-14)*x)*P(n-2,x) + (4*n-14)*x*d/dx P(n-2,x) - (7*n-12+(4*n-6)*x)*P(n-1,x) - (4*n-6)*x*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 5 + 11*x + 3*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 5 + 11*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 4)*P[n - 3, x] + (7*n - 23 + (4*n - 14)*x)*P[n - 2, x] + (4*n - 14)*x*D[P[n - 2, x], x] - (7*n - 12 + (4*n - 6)*x)*P[n - 1, x] - (4*n - 6)*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] = 1 + x; v[3] = 5 + 11*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-(n - 4)*v[i - 3] + (7*n - 23 + (4*n - 14)*x)*v[i - 2] + (4*n - 14)*x*deriv(v[i - 2], x) - (7*n - 12 + (4*n - 6)*x)*v[i - 1] - (4*n - 6)*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

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.