Real-rooted atlas

A085881 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
22721351
3405555180151155
485051837510185189010539060
5229635738045623700193725236259451809675
67577955349999654242199519646550395010034303510395108949995
72955402451918511595321242922020839168356148642508702694056756701351358218099890

All 15 rows

Technical details
Node id
n-f67f18b926dba7e3 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-322b0642acfc2857

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:            3,             1;
  2:           27,            21,              3;
  3:          405,           555,            180,             15;
  4:         8505,         18375,          10185,           1890,            105;
  5:       229635,        738045,         623700,         193725,          23625,           945;
  6:      7577955,      34999965,       42421995,       19646550,        3950100,        343035,        10395;
  7:    295540245,    1918511595,     3212429220,     2083916835,      614864250,      87026940,      5675670,      135135;
  8:  13299311025,  119543799375,   269709865425,   235738953450,    95819498775,   19666196550,   2071619550,   105405300,    2027025;
  9: 678264862275, 8355066644925, 24957583150500, 28630440964125, 15410082562875, 4303534209975, 651283132500, 53136433350, 2170943775, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (6*n-3+(2*n-1)*x)*P(n-1,x) + (2*n-1)*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[(6*n - 3 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*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] = (6*n - 3 + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*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.