Real-rooted atlas

A034870 → x ↦ x+1 → basis A013610

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 18\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1918936
281324486324811296
37294374109351458010935437472946656
46561524881837083674164592703674161837085248865611679616
55904959049026572057085880124002901488034812400290708588026572055904905904960466176
65314416377292350751061169170202630632954209012724910514844209012722630632951169170203507510663772925314412176782336
747829696696156643525017917410007164787751969957550393814363255907164151496081436325590795755039384787751969174100071643525017966961566478296978364164096

All 15 rows

Technical details
Node id
n-9335dcc335de4078 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-2cd9f46ca2d53115

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: yes
  • γ-positive (gamma_positive): yes
  • 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:         9,         18,           9;
  2:        81,        324,         486,          324,            81;
  3:       729,       4374,       10935,        14580,         10935,          4374,           729;
  4:      6561,      52488,      183708,       367416,        459270,        367416,        183708,          52488,           6561;
  5:     59049,     590490,     2657205,      7085880,      12400290,      14880348,      12400290,        7085880,        2657205,         590490,          59049;
  6:    531441,    6377292,    35075106,    116917020,     263063295,     420901272,     491051484,      420901272,      263063295,      116917020,       35075106,        6377292,        531441;
  7:   4782969,   66961566,   435250179,   1741000716,    4787751969,    9575503938,   14363255907,    16415149608,    14363255907,     9575503938,     4787751969,     1741000716,     435250179,      66961566,       4782969;
  8:  43046721,  688747536,  5165606520,  24106163760,   78345032220,  188028077328,  344718141768,   492454488240,   554011299270,   492454488240,   344718141768,   188028077328,   78345032220,   24106163760,    5165606520,    688747536,    43046721;
  9: 387420489, 6973568802, 59275334817, 316135119024, 1185506696340, 3319418749752, 7192073957796, 12329269641936, 16952745757662, 18836384175180, 16952745757662, 12329269641936, 7192073957796, 3319418749752, 1185506696340, 316135119024, 59275334817, 6973568802, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (9+18*x+9*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[(9 + 18*x + 9*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] = (9 + 18*x + 9*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.