Real-rooted atlas

basis-map coefficients of A026729 → basis A013611 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
251621
325160256441
41251200384040969261
56258000384008192065536194481
63125500003200001024000163840010485764084101
71562530000024000001024000024576000314572801677721685766121
878125175000016800000896000002867200005505024005872025602684354561801088541
939062510000000112000000716800000286720000073400320001174405120010737418240429496729637822859361
101953125562500007200000005376000000258048000008257536000017616076800024159191040019327352832068719476736794280046581
1197656253125000004500000000384000000002150400000008257536000002202009600000402653184000048318382080003435973836800109951162777616679880978201
124882812517187500002750000000026400000000016896000000007569408000000242221056000005536481280000088583700480000944892805120006047313952768017592186044416350277500542221
1324414062593750000001650000000001760000000000126720000000006488064000000024222105600000066437775360000013287555072000001889785610240000181419418583040010555311626649602814749767106567355827511386641
1412207031255078125000097500000000011440000000000915200000000005271552000000002249195520000000719742566400000017273821593600000307090161664000003930754069299200034304762786611200182958734861926404503599627370496154472377739119461
15610351562527343750000056875000000007280000000000064064000000000041000960000000001968046080000000071974256640000000201527918592000000429926226329600000687881962127360000800444465020928000640355572016742400315251973915934720720575940379279363243919932521508681
Technical details
Node id
n-5b2c6b7db5434b28 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-833b492313a93e6e

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       5,       16;
   3:      25,      160,       256;
   4:     125,     1200,      3840,       4096;
   5:     625,     8000,     38400,      81920,       65536;
   6:    3125,    50000,    320000,    1024000,     1638400,     1048576;
   7:   15625,   300000,   2400000,   10240000,    24576000,    31457280,     16777216;
   8:   78125,  1750000,  16800000,   89600000,   286720000,   550502400,    587202560,    268435456;
   9:  390625, 10000000, 112000000,  716800000,  2867200000,  7340032000,  11744051200,  10737418240,   4294967296;
  10: 1953125, 56250000, 720000000, 5376000000, 25804800000, 82575360000, 176160768000, 241591910400, 193273528320, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (5+16*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); 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, v[i])))};
T(10)

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