Real-rooted atlas

A038334 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(20 + 48\,x + 28\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1120\,x + 784\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
040011207842304
18000336004704021952110592
2160000896000188160017561606146565308416
332000002240000062720000878080006146560017210368254803968
464000000537600000188160000035123200003687936000206524416048189030412230590464
5128000000012544000000526848000001229312000001721036800001445670912006746464256013492928512587068342272
62560000000028672000000014049280000003933798400000688414720000077102448640005397171404800215886856192037780199833628179280429056
751200000000064512000000003612672000000011801395200000024782929920000034696101888000032383028428800019429817057280068004359700480105784559534081352605460594688

All 15 rows

Technical details
Node id
n-dd8520417c7f5dfb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f96e0317bce1f626

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             400,             1120,               784;
  1:            8000,            33600,             47040,             21952;
  2:          160000,           896000,           1881600,           1756160,             614656;
  3:         3200000,         22400000,          62720000,          87808000,           61465600,           17210368;
  4:        64000000,        537600000,        1881600000,        3512320000,         3687936000,         2065244160,          481890304;
  5:      1280000000,      12544000000,       52684800000,      122931200000,       172103680000,       144567091200,        67464642560,        13492928512;
  6:     25600000000,     286720000000,     1404928000000,     3933798400000,      6884147200000,      7710244864000,      5397171404800,      2158868561920,       377801998336;
  7:    512000000000,    6451200000000,    36126720000000,   118013952000000,    247829299200000,    346961018880000,    323830284288000,    194298170572800,     68004359700480,     10578455953408;
  8:  10240000000000,  143360000000000,   903168000000000,  3371827200000000,   8260976640000000,  13878440755200000,  16191514214400000,  12953211371520000,   6800435970048000,   2115691190681600,   296196766695424;
  9: 204800000000000, 3153920000000000, 22077440000000000, 92725248000000000, 259630694400000000, 508876161024000000, 712426625433600000, 712426625433600000, 498698637803520000, 232726030974976000, 65163288672993280, 8293509467471872;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (20+48*x+28*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1120*x + 784*x^2.
%t Mathematica
P[0, x_] := 400 + 1120*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((20 + 48*x + 28*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 1120*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((20 + 48*x + 28*x^2)*v[i - 1])/(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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.