Real-rooted atlas

A038333 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(47 + 28\,x + 28\,n\right)\,P_{n}(x) = \left(1425 + 2632\,x + 784\,x^{2} + 532\,n + 784\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1425 + 2632\,x + 784\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0142526327844841
13718311970011054421952289379
289852941644966703200412697661465616507857
32072103912579406029151472031281600014444416017210368912500347
44630305133481134552105667010401632482432013138272000485332377648189030449309176505
51011486441590753980548341151186096690357801280799916391680515020262400158541910016134929285122619349324947
6217210832577226572962896010164445821376254726218951683866003687168035836254347264192274231296005073341120512377801998336137294865645473
7460254558411154737129809404285481933248960853813448995584160477517939558419484658583326721505122682585088692187232665600159810245296128105784559534087119574711866539

All 15 rows

Technical details
Node id
n-d34ea40cdf30ed64 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a55687d918dbb23b

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:             1425,              2632,                784;
  1:            37183,            119700,             110544,              21952;
  2:           898529,           4164496,            6703200,            4126976,              614656;
  3:         20721039,         125794060,          291514720,          312816000,           144444160,            17210368;
  4:        463030513,        3481134552,        10566701040,        16324824320,         13138272000,          4853323776,           481890304;
  5:      10114864415,       90753980548,       341151186096,       690357801280,        799916391680,        515020262400,        158541910016,         13492928512;
  6:     217210832577,     2265729628960,     10164445821376,     25472621895168,      38660036871680,      35836254347264,      19227423129600,       5073341120512,        377801998336;
  7:    4602545584111,    54737129809404,    285481933248960,    853813448995584,    1604775179395584,    1948465858332672,    1505122682585088,     692187232665600,     159810245296128,     10578455953408;
  8:   96483621635921,  1288712763551080,   7663198173316560,  26644980436569600,   59766941429690880,   89867410046152704,   90928406722191360,   60204907303403520,   24226553143296000,   4971874298101760,    296196766695424;
  9: 2004858666300927, 29716955463863668, 198461765586866320, 786755012460500160, 2051663493615859200, 3681643592068958208, 4613193715702505472, 4000849895776419840, 2317888931181035520, 829086485348352000, 153133728381534208, 8293509467471872;
%F Formula
The row polynomials satisfy (28*n+47+28*x)*P(n,x) = (532*n+1425+(784*n+2632)*x+784*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1425 + 2632*x + 784*x^2.
%t Mathematica
P[0, x_] := 1425 + 2632*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((532*n + 1425 + (784*n + 2632)*x + 784*x^2)*P[n - 1, x])/(28*n + 47 + 28*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1425 + 2632*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((532*n + 1425 + (784*n + 2632)*x + 784*x^2)*v[i - 1])/(28*n + 47 + 28*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

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.