Real-rooted atlas

basis-map coefficients of A026729 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2549
325561697
4125604432641225
562559368032281625617665
631255592412816080000166401024284873
715625515816188977618726406694409216040965059553
878125470484426630352393939842154880049817604874241638497841673
939062542656096365013952776219136610100736210739200340623362490368655362041737985
10195312538546736449158057601465578112015916992000770888294418307645442184314881238630426214445646726793
119765625347701877665447344336268811502080392879956480256390041600839277035521453129728013320192006029312010485761086867990625
12488281253133223148486472355147248315849127049325725094400798127086592034594635694088152436244481074098995207800094720288358400419430427424895224905
13244140625282185395856113667350950728559583772185621516656663321623683467202560013224952697651240740893884416723514078003274944348160044176506880135895449616777216730266798373377
14122070312525406451252041488962978193601500596181651840486088125018496067815330671728644782574970421248187344760609177643066378439884859667745865728498571673600024329479782463250104326710886420446038173088073
156103515625228706889393361945814452152496261057202793587201080808909804915201889818516800614401658208064109076488095407088271360023428350447124480416903271809024046302823658291231864312037376130840474419229125246976268435456600005644990402017
Technical details
Node id
n-57992f4258de839d · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c7f57c7d3a795bc5

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:       5,         4;
   3:      25,        56,         16;
   4:     125,       604,        432,          64;
   5:     625,      5936,       8032,        2816,         256;
   6:    3125,     55924,     128160,       80000,       16640,       1024;
   7:   15625,    515816,    1889776,     1872640,      669440,      92160,       4096;
   8:   78125,   4704844,   26630352,    39393984,    21548800,    4981760,     487424,     16384;
   9:  390625,  42656096,  365013952,   776219136,   610100736,  210739200,   34062336,   2490368,    65536;
  10: 1953125, 385467364, 4915805760, 14655781120, 15916992000, 7708882944, 1830764544, 218431488, 12386304, 262144;
%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) = 1.
%t Mathematica
P[1, x_] := 1;
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] = 1; 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.