Real-rooted atlas

A200536 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(15 + 13\,x + 2\,x^{2}\right)\,P_{n-1}(x) + \left(13\,x + 4\,x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11513230
22255593416841197
333751857725298127672746252863023
4506255640311460969141103461965713540815032800164257572
575937516450993754333821249366559458657537041081800767497900066560232032358263647
61139062547049967936756878219862822238118816614647437813705263670294555637610071529708559352025720063366436610341405
7170859375133220691371734272277387290767113071335153124816125683147153267262672577821814509537544608188538587540058050073189627213648903392165761284449765739816

All 15 rows

Technical details
Node id
n-29feaac19fa9e41e · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-669843b93c09642d

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:          15,             13,               2;
  2:         225,            559,             341,               68,                 4;
  3:        3375,          18577,           25298,            12767,              2746,               252,                 8;
  4:       50625,         564031,         1460969,          1411034,            619657,            135408,             15032,               800,               16;
  5:      759375,       16450993,        75433382,        124936655,          94586575,          37041081,           8007674,            979000,            66560,             2320,              32;
  6:    11390625,      470499679,      3675687821,       9862822238,       11881661464,        7437813705,        2636702945,         556376100,         71529708,          5593520,          257200,           6336,            64;
  7:   170859375,    13322069137,    173427227738,     729076711307,     1335153124816,     1256831471532,      672626725778,      218145095375,      44608188538,       5875400580,       500731896,       27213648,        903392,        16576,         128;
  8:  2562890625,   375239107711,   8031042142289,   51787678867034,   140072706596407,   190997370626118,   147057633854222,    68848419918100,   20593693914209,    4065269235040,    539848871696,    48576350592,    2945617248,    117755904,     2956032,     41984,     256;
  9: 38443359375, 10540012594033, 367781087015342, 3587389168427735, 14057370356233465, 27085368324164031, 29082876623550674, 18904326978297450, 7874043778532823, 2186427374846645, 415836463646578, 55104389193744, 5129893230016, 335242281696, 15203702208, 465927936, 9156864, 103680, 512;
%F Formula
The row polynomials satisfy P(n,x) = (15+13*x+2*x^2)*P(n-1,x) + (13*x+4*x^2)*d/dx P(n-1,x) + 2*x^2*d^2/dx^2 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[(15 + 13*x + 2*x^2)*P[n - 1, x] + (13*x + 4*x^2)*D[P[n - 1, x], x] + 2*x^2*D[P[n - 1, x], {x, 2}]];
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] = (15 + 13*x + 2*x^2)*v[i - 1] + (13*x + 4*x^2)*deriv(v[i - 1], x) + 2*x^2*deriv(deriv(v[i - 1], x), 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

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.