Real-rooted atlas

A139548 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 24\,x + 8\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11824850
2324864864384642500
35832233283888034560172804608512125000
41049765598721306368174182414515207741442580484915240966250000
5188956812597120377913606718464078382080627056643483648013271040331776049152032768312500000
6340122242720977929976919042217093120332563968035473489922759049216157659955265691648019464192038928384471859226214415625000000
7612220032571405363224760899072660290641921210532843521614043791361614043791361229747650567173527961631882346496106274488322576351232429391872440401922097152781250000000

All 15 rows

Technical details
Node id
n-af48e57b5ab0efc6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6b852c5cd7d4c499

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:            1;
  1:           18,            24,              8;
  2:          324,           864,            864,            384,              64;
  3:         5832,         23328,          38880,          34560,           17280,            4608,             512;
  4:       104976,        559872,        1306368,        1741824,         1451520,          774144,          258048,           49152,            4096;
  5:      1889568,      12597120,       37791360,       67184640,        78382080,        62705664,        34836480,        13271040,         3317760,          491520,           32768;
  6:     34012224,     272097792,      997691904,     2217093120,      3325639680,      3547348992,      2759049216,      1576599552,       656916480,       194641920,        38928384,        4718592,         262144;
  7:    612220032,    5714053632,    24760899072,    66029064192,    121053284352,    161404379136,    161404379136,    122974765056,     71735279616,     31882346496,     10627448832,     2576351232,      429391872,      44040192,       2097152;
  8:  11019960576,  117546246144,   587731230720,  1828497162240,   3961743851520,   6338790162432,   7747410198528,   7378485903360,   5533864427520,   3279327068160,   1530352631808,   556491866112,   154581073920,   31708938240,    4529848320,    402653184,    16777216;
  9: 198359290368, 2380311484416, 13488431745024, 47958868426752, 119897171066880, 223808052658176, 323278298284032, 369460912324608, 338672502964224, 250868520714240, 150521112428544, 72979933298688, 28381085171712, 8732641591296, 2079200378880, 369635622912, 46204452864, 3623878656, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (18+24*x+8*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(18 + 24*x + 8*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (18 + 24*x + 8*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.