Real-rooted atlas

basis-map coefficients of A112227 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 25 + 56\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
125561697
2125604432641225
362559368032281625617665
431255592412816080000166401024284873
515625515816188977618726406694409216040965059553
678125470484426630352393939842154880049817604874241638497841673
739062542656096365013952776219136610100736210739200340623362490368655362041737985
8195312538546736449158057601465578112015916992000770888294418307645442184314881238630426214445646726793

All 15 rows

Technical details
Node id
n-33e1acaf6d61c325 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c7f57c7d3a795bc5

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:       25,          56,           16;
   2:      125,         604,          432,            64;
   3:      625,        5936,         8032,          2816,           256;
   4:     3125,       55924,       128160,         80000,         16640,          1024;
   5:    15625,      515816,      1889776,       1872640,        669440,         92160,          4096;
   6:    78125,     4704844,     26630352,      39393984,      21548800,       4981760,        487424,        16384;
   7:   390625,    42656096,    365013952,     776219136,     610100736,     210739200,      34062336,      2490368,        65536;
   8:  1953125,   385467364,   4915805760,   14655781120,   15916992000,    7708882944,    1830764544,    218431488,     12386304,     262144;
   9:  9765625,  3477018776,  65447344336,  268811502080,  392879956480,  256390041600,   83927703552,  14531297280,   1332019200,   60293120,   1048576;
  10: 48828125, 31332231484, 864723551472, 4831584912704, 9325725094400, 7981270865920, 3459463569408, 815243624448, 107409899520, 7800094720, 288358400, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (5+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 25 + 56*x + 16*x^2.
%t Mathematica
P[1, x_] := 25 + 56*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 25 + 56*x + 16*x^2; for(n = 2, N, my(i = n); v[i] = (5 + 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, 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.