Real-rooted atlas

A196347 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
23240880
3384912432481776
461442496021120537638457984
512288081024010944004800007680038402588160
6294912030643200622540804193280011289600124416046080150359040
78257536013283020803915233280380685312015353856002799820802257920064512010971555840

All 15 rows

Technical details
Node id
n-9392aab10f8a4259 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

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:           4,             2;
  2:          32,            40,              8;
  3:         384,           912,            432,             48;
  4:        6144,         24960,          21120,           5376,            384;
  5:      122880,        810240,        1094400,         480000,          76800,           3840;
  6:     2949120,      30643200,       62254080,       41932800,       11289600,        1244160,         46080;
  7:    82575360,    1328302080,     3915233280,     3806853120,     1535385600,      279982080,      22579200,       645120;
  8:  2642411520,   65079705600,   271827763200,   367191982080,   208306667520,    55924162560,    7369850880,    454164480,    10321920;
  9: 95126814720, 3561867509760, 20743033651200, 37940178124800, 29106575769600, 10795964497920, 2067893452800, 206231961600, 10032906240, 185794560;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+2*n*x)*P(n-1,x) + 2*n*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n + 2*n*x)*P[n - 1, x] + 2*n*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n + 2*n*x)*v[i - 1] + 2*n*x*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 - 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

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.