Real-rooted atlas

A038319 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 25 + 160\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
025160256441
11251200384040969261
26258000384008192065536194481
33125500003200001024000163840010485764084101
41562530000024000001024000024576000314572801677721685766121
578125175000016800000896000002867200005505024005872025602684354561801088541
639062510000000112000000716800000286720000073400320001174405120010737418240429496729637822859361
71953125562500007200000005376000000258048000008257536000017616076800024159191040019327352832068719476736794280046581
897656253125000004500000000384000000002150400000008257536000002202009600000402653184000048318382080003435973836800109951162777616679880978201
94882812517187500002750000000026400000000016896000000007569408000000242221056000005536481280000088583700480000944892805120006047313952768017592186044416350277500542221
1024414062593750000001650000000001760000000000126720000000006488064000000024222105600000066437775360000013287555072000001889785610240000181419418583040010555311626649602814749767106567355827511386641
1112207031255078125000097500000000011440000000000915200000000005271552000000002249195520000000719742566400000017273821593600000307090161664000003930754069299200034304762786611200182958734861926404503599627370496154472377739119461
12610351562527343750000056875000000007280000000000064064000000000041000960000000001968046080000000071974256640000000201527918592000000429926226329600000687881962127360000800444465020928000640355572016742400315251973915934720720575940379279363243919932521508681
133051757812514648437500003281250000000045500000000000043680000000000003075072000000000016400384000000000067475865600000000021592276992000000005374077829120000000103182294319104000001500833371914240000016008889300418560000118219490218475520005404319552844595200115292150460684697668122318582951682301
14152587890625781250000000018750000000000028000000000000002912000000000000022364160000000000013120307200000000005997854720000000000215922769920000000006141803233280000000013757639242547200000024013333950627840000032017778600837120000031525197391593472000021617278211378380800092233720368547758080184467440737095516161430568690241985328321
Technical details
Node id
n-8bfae0a3670d90bd · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-833b492313a93e6e

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,        160,         256;
  1:      125,       1200,        3840,         4096;
  2:      625,       8000,       38400,        81920,         65536;
  3:     3125,      50000,      320000,      1024000,       1638400,       1048576;
  4:    15625,     300000,     2400000,     10240000,      24576000,      31457280,       16777216;
  5:    78125,    1750000,    16800000,     89600000,     286720000,     550502400,      587202560,      268435456;
  6:   390625,   10000000,   112000000,    716800000,    2867200000,    7340032000,    11744051200,    10737418240,     4294967296;
  7:  1953125,   56250000,   720000000,   5376000000,   25804800000,   82575360000,   176160768000,   241591910400,   193273528320,    68719476736;
  8:  9765625,  312500000,  4500000000,  38400000000,  215040000000,  825753600000,  2202009600000,  4026531840000,  4831838208000,  3435973836800,  1099511627776;
  9: 48828125, 1718750000, 27500000000, 264000000000, 1689600000000, 7569408000000, 24222105600000, 55364812800000, 88583700480000, 94489280512000, 60473139527680, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = (5+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 160*x + 256*x^2.
%t Mathematica
P[0, x_] := 25 + 160*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 16*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 + 160*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 16*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.