Real-rooted atlas

A028326 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
110616
2507818146
3250774432541510
4125069427074205216217480
562505928698640499508910486223522
63125049303812628989952203013203645014583121634
71562504038054153709921772177481081001632960142884437447175388

All 15 rows

Technical details
Node id
n-2ba46ce99cd418a9 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c377b8329dc9dd7c

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:       2;
  1:      10,         6;
  2:      50,        78,         18;
  3:     250,       774,        432,         54;
  4:    1250,      6942,       7074,       2052,        162;
  5:    6250,     59286,      98640,      49950,       8910,        486;
  6:   31250,    493038,    1262898,     995220,     301320,      36450,      1458;
  7:  156250,   4038054,   15370992,   17721774,    8108100,    1632960,    142884,     4374;
  8:  781250,  32773182,  181195074,  294217812,  191003022,   56983500,   8185212,   542376,   13122;
  9: 3906250, 264529206, 2091465360, 4662634590, 4129704810, 1712679066, 359210376, 38657412, 2007666, 39366;
%F Formula
The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(5 + 3*x)*P[n - 1, x] + 3*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] = 2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 3*x)*v[i - 1] + 3*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.