Real-rooted atlas

A038284 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(20 + 6\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 400 + 276\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
040027636712
180009576280821620600
2160000296976147312250561296630640
33200000868137664958401836000207360777620428352
46400000024491577625995513610874304020139840163296046656699433408
5128000000067518101769788059008569196633615386112002024870401241049627993625265624192
625600000000183227064576353728749312275023074816101850690816193560192001909910016918190081679616960789007360
75120000000004917503678976124186823654401257324933888061315688448001578905104896223091076096171522385926651279361007769638372827853312

All 15 rows

Technical details
Node id
n-cf478dba13e9a5b6 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-277523cbbaaedfaf

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:             400,              276,                36;
  1:            8000,             9576,              2808,               216;
  2:          160000,           296976,            147312,             25056,              1296;
  3:         3200000,          8681376,           6495840,           1836000,            207360,             7776;
  4:        64000000,        244915776,         259955136,         108743040,          20139840,          1632960,            46656;
  5:      1280000000,       6751810176,        9788059008,        5691966336,        1538611200,        202487040,         12410496,          279936;
  6:     25600000000,     183227064576,      353728749312,      275023074816,      101850690816,      19356019200,       1909910016,        91819008,        1679616;
  7:    512000000000,    4917503678976,    12418682365440,    12573249338880,     6131568844800,    1578905104896,     223091076096,     17152238592,      665127936,      10077696;
  8:  10240000000000,  130927095653376,   426902857767936,   552295569070080,   345228525204480,  115734668313600,   21966530890752,   2401985249280,   148142131200,    4736517120,    60466176;
  9: 204800000000000, 3465544486987776, 14446454022494208, 23548648771270656, 18503828523417600, 7858104566906880, 1924533739763712, 280722270799872, 24485576417280, 1239355054080, 33256396800, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = (20+6*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 400 + 276*x + 36*x^2.
%t Mathematica
P[0, x_] := 400 + 276*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 6*x)*P[n - 1, x] + 6*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] = 400 + 276*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 6*x)*v[i - 1] + 6*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.