Real-rooted atlas

A120910 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
1211233
214721648411
31029296415841925769
472033672035616998476890291
5504214327326811203321605760030721557105
6352947496173611947728903552026534403133441228829277003
72470629559908841990628642194657929717120019074048163430449152594918873

All 15 rows

Technical details
Node id
n-6c7b3ea8bb66c042 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-01a7a97b7ee8f4b3

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:         3;
  1:        21,         12;
  2:       147,        216,          48;
  3:      1029,       2964,        1584,          192;
  4:      7203,      36720,       35616,         9984,         768;
  5:     50421,     432732,      681120,       332160,       57600,        3072;
  6:    352947,    4961736,    11947728,      9035520,     2653440,      313344,      12288;
  7:   2470629,   55990884,   199062864,    219465792,    97171200,    19074048,    1634304,     49152;
  8:  17294403,  625782240,  3209906496,   4966101504,  3112800768,   903684096,  126959616,   8257536,   196608;
  9: 121060821, 6952782252, 50651726400, 107195554560, 91458823680, 36850673664, 7550484480, 796852224, 40697856, 786432;
%F Formula
The row polynomials satisfy P(n,x) = (7+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(7 + 4*x)*P[n - 1, x] + 4*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] = 3; for(n = 1, N, my(i = n + 1); v[i] = (7 + 4*x)*v[i - 1] + 4*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.