Real-rooted atlas

A038333 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(40 + 21\,x\right)\,P_{n-1}(x) + \left(21\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1600 + 2121\,x + 441\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0160021214414162
164000162981807039261316945
225600001128584110040247264864619448126729215
310240000074219630110603029154836557257973721040841012472376252
44096000000474243743611025309613517208290089020044184265226667605585766121248530863043
5163840000000297890283602194034506923639577688980041399921976755074959589754061837373241180108854126936336635297
665536000000001851537129972818336399163302071183974429483846697034719757061192675020261850260065178063461635388395228378228593613126711127891180
7262144000000000114320020928341417224670111201987513945580447977048511129576826903633042575607052866531836325738135228685195450384090242210311046876794280046581386526243482814007

All 15 rows

Technical details
Node id
n-4373fb8f079b57b6 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-bb53eddc79186abf

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 2)
  • 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:               1600,                 2121,                   441;
  1:              64000,               162981,                 80703,                   9261;
  2:            2560000,             11285841,              10040247,                2648646,                 194481;
  3:          102400000,            742196301,            1060302915,              483655725,               79737210,                4084101;
  4:         4096000000,          47424374361,          102530961351,            72082900890,            20044184265,             2266676055,              85766121;
  5:       163840000000,        2978902836021,         9403450692363,          9577688980041,          3999219767550,           749595897540,           61837373241,            1801088541;
  6:      6553600000000,      185153712997281,       833639916330207,       1183974429483846,        697034719757061,        192675020261850,        26006517806346,         1635388395228,          37822859361;
  7:    262144000000000,    11432002092834141,     72246701112019875,     139455804479770485,     111295768269036330,      42575607052866531,      8363257381352286,       851954503840902,       42210311046876,       794280046581;
  8:  10485760000000000,   702857151662882601,   6164301535135146711,   15881128584768777330,   16729247159435685105,    8510674156315409925,   2282388473414676627,    334943897226646680,    26670789278409150,   1068306662651445,    16679880978201;
  9: 419430400000000000, 43094487211435838661, 520232726066002564923, 1765206576469022145921, 2407930348050169276950, 1585361943013883826330, 557600643869459928507, 110564666723091138327, 12581346011668683480, 804728800593773055, 26604410160230595, 350277500542221;
%F Formula
The row polynomials satisfy P(n,x) = (40+21*x)*P(n-1,x) + 21*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1600 + 2121*x + 441*x^2.
%t Mathematica
P[0, x_] := 1600 + 2121*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[(40 + 21*x)*P[n - 1, x] + 21*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] = 1600 + 2121*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = (40 + 21*x)*v[i - 1] + 21*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

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.