Real-rooted atlas

A085880 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-36 - 24\,x + 24\,n + 16\,n x\right)\,P_{n-1}(x) + \left(-24\,x + 16\,n x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26410
37212832232
41080392024003207720
518144121856137984430083584324576
63265923873408725760039782407526404300816231488
7615859212584140836802867231629312010205184012976128540672931890432
812009254441699074561838124288023287520256114085171202423156736221405184702873660018870912

All 15 rows

Technical details
Node id
n-cae3750b8f2fc5bd · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

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:           6,             4;
   3:          72,           128,             32;
   4:        1080,          3920,           2400,             320;
   5:       18144,        121856,         137984,           43008,            3584;
   6:      326592,       3873408,        7257600,         3978240,          752640,          43008;
   7:     6158592,     125841408,      368028672,       316293120,       102051840,       12976128,         540672;
   8:   120092544,    4169907456,    18381242880,     23287520256,     11408517120,     2423156736,      221405184,       7028736;
   9:  2401850880,  140598149120,   913390100480,   1642334453760,   1147124858880,   362120478720,    54449274880,    3748659200,    93716480;
  10: 48997757952, 4813002242048, 45389503610880, 112932973936640, 108140688015360, 47612348399616, 10478664548352, 1173855141888, 63089934336, 1274544128;
%F Formula
The row polynomials satisfy n*P(n,x) = (24*n-36+(16*n-24)*x)*P(n-1,x) + (16*n-24)*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((24*n - 36 + (16*n - 24)*x)*P[n - 1, x] + (16*n - 24)*x*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((24*n - 36 + (16*n - 24)*x)*v[i - 1] + (16*n - 24)*x*deriv(v[i - 1], x))/(n)); 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

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.