Real-rooted atlas

A038331 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 17\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 680\,x + 289\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04006802891369
180002040017340491350653
2160000544000693600393040835211874161
332000001360000023120000196520008352100141985769343957
464000000326400000693600000786080000501126000170382840241375692565726409
51280000000761600000019420800000275128000002338588000011926798800337925966041033867394931877133
6256000000001740800000005178880000008804096000009354352000006360959360002703407728006565418768069757574413512479453921
75120000000003916800000000133171200000002641228800000033675667200000286243171200001622044636800059088768912001255636339380118587876497129961739795077

All 15 rows

Technical details
Node id
n-2405be3a93998241 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8c1ae502a0a48e51

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,              680,              289;
  1:            8000,            20400,            17340,              4913;
  2:          160000,           544000,           693600,            393040,             83521;
  3:         3200000,         13600000,         23120000,          19652000,           8352100,           1419857;
  4:        64000000,        326400000,        693600000,         786080000,         501126000,         170382840,          24137569;
  5:      1280000000,       7616000000,      19420800000,       27512800000,       23385880000,       11926798800,        3379259660,         410338673;
  6:     25600000000,     174080000000,     517888000000,      880409600000,      935435200000,      636095936000,      270340772800,       65654187680,       6975757441;
  7:    512000000000,    3916800000000,   13317120000000,    26412288000000,    33675667200000,    28624317120000,    16220446368000,     5908876891200,    1255636339380,     118587876497;
  8:  10240000000000,   87040000000000,  332928000000000,   754636800000000,  1122522240000000,  1144972684800000,   811022318400000,   393925126080000,  125563633938000,   23717575299400,   2015993900449;
  9: 204800000000000, 1914880000000000, 8138240000000000, 20752512000000000, 35279270400000000, 41982331776000000, 35684982009600000, 21665881934400000, 9207999822120000, 2608933282934000, 443518658098780, 34271896307633;
%F Formula
The row polynomials satisfy P(n,x) = (20+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 680*x + 289*x^2.
%t Mathematica
P[0, x_] := 400 + 680*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 17*x)*P[n - 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 + 680*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 17*x)*v[i - 1]); 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.