Real-rooted atlas

A159854 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 25 + 80\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0258064169
115068010245122366
290052801158411264409633124
354003888011174416025611468832768463736
4324002764809815041855488197017611141122621446492304
51944001918080810086418984960266649602244608010485760209715290892256
61166400130636806394982417871667231186944034799616024248320096468992167772161272491584
76998400877132804882083841583898624330095001645829324804238868480251867955287241523213421772817814882176

All 15 rows

Technical details
Node id
n-2f46ab61d8cae267 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-249fd621b3e0dd99

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:        25,         80,          64;
  1:       150,        680,        1024,          512;
  2:       900,       5280,       11584,        11264,         4096;
  3:      5400,      38880,      111744,       160256,       114688,        32768;
  4:     32400,     276480,      981504,      1855488,      1970176,      1114112,       262144;
  5:    194400,    1918080,     8100864,     18984960,     26664960,     22446080,     10485760,      2097152;
  6:   1166400,   13063680,    63949824,    178716672,    311869440,    347996160,    242483200,     96468992,     16777216;
  7:   6998400,   87713280,   488208384,   1583898624,   3300950016,   4582932480,   4238868480,   2518679552,    872415232,    134217728;
  8:  41990400,  582266880,  3630956544,  13409058816,  32476889088,  53905195008,  62096670720,  49023025152,  25383927808,   7784628224,  1073741824;
  9: 251942400, 3829524480, 26443874304, 109502005248, 302133805056, 583246282752, 803821584384, 790911516672, 544487768064, 249779191808, 68719476736, 8589934592;
%F Formula
The row polynomials satisfy P(n,x) = (6+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 80*x + 64*x^2.
%t Mathematica
P[0, x_] := 25 + 80*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 8*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] = 25 + 80*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 8*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

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