Real-rooted atlas

A024462 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-8 - 6\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-6\,x + 3\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 4 + 5\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
245110
3164725391
4643773911149955
52562831504126554862711296
6102420585589034963815741197181147943
740961471676507858220034007708467276952432117431

All 15 rows

Technical details
Node id
n-b8827d5bb870ce1f · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-fb9a371d3330523a

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     2,       1;
  2:     4,       5,        1;
  3:    16,      47,       25,         3;
  4:    64,     377,      391,       114,         9;
  5:   256,    2831,     5041,      2655,       486,       27;
  6:  1024,   20585,    58903,     49638,     15741,     1971,       81;
  7:  4096,  147167,   650785,    822003,    400770,    84672,     7695,     243;
  8: 16384, 1042457,  6949351,  12638394,   8878329,  2811078,   423306,   29160,    729;
  9: 65536, 7346351, 72620881, 185147175, 179968446, 80045469, 17745966, 1998918, 107892, 2187;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(3*n-6)*x)*P(n-1,x) + (3*n-6)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 4 + 5*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 4 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (3*n - 6)*x)*P[n - 1, x] + (3*n - 6)*x*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 4 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (3*n - 6)*x)*v[i - 1] + (3*n - 6)*x*deriv(v[i - 1], x))/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

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

Proof chains ? JSON

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.