Real-rooted atlas

A038330 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(20 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 400 + 176\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
040017616592
18000582411526415040
2160000171776555526656256394240
332000004762624224256043520035840102410677248
464000000127102976818421762289664030310401843204096299061248
5128000000033064714242799992832106006118420070400019496960917504163848667660288
625600000000844753141769162568499245121929216114655887361582694400118358016445644865536259994091520
751200000000021298075402242903420436480181040447488059324891136010917013094411538530304687341568212336642621448070298861568

All 15 rows

Technical details
Node id
n-19d88b15847dfa77 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-73b18cc810b5ad72

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,              176,               16;
  1:            8000,             5824,             1152,               64;
  2:          160000,           171776,            55552,             6656,              256;
  3:         3200000,          4762624,          2242560,           435200,            35840,            1024;
  4:        64000000,        127102976,         81842176,         22896640,          3031040,          184320,           4096;
  5:      1280000000,       3306471424,       2799992832,       1060061184,        200704000,        19496960,         917504,         16384;
  6:     25600000000,      84475314176,      91625684992,      45121929216,      11465588736,      1582694400,      118358016,       4456448,        65536;
  7:    512000000000,    2129807540224,    2903420436480,    1810404474880,     593248911360,    109170130944,    11538530304,     687341568,     21233664,      262144;
  8:  10240000000000,   53163380965376,   89815002382336,   69546624942080,   28598578708480,   6739800883200,   944375857152,   79146516480,   3853516800,    99614720,   1048576;
  9: 204800000000000, 1316881143169024, 2727473590566912, 2584752007675904, 1307735333273600, 383986350161920, 68511741247488, 7576536219648, 516968939520, 20992491520, 461373440, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (20+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 400 + 176*x + 16*x^2.
%t Mathematica
P[0, x_] := 400 + 176*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 4*x)*P[n - 1, x] + 4*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 + 176*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 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 - 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.