Real-rooted atlas

A130534 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9\,x^{2} - 6\,n x - n^{2}\right)\,P_{n-2}(x) + \left(1 + 6\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2151833
37816281321
4462127811343243198
5313210440125556480121533822
6240849187213486595580328054374383580
720779287948014923441304100620865153090153094672980
81991088916207217388756177062761047816036129246735965248861065360
92099952010356724821511688424661778417118013573850616193811942834352177147853724880
1024188112012652368482832338340357326316028077981751428605262471670290976276801151455559049012730525920
1130223238401663412889639681625896542085112804709331495027251336574106601618972788687440467668080454677301948617201855175200
12407272147202343776186885907147166808634371502248154950116505238033092822339617826707280206322415491505150214997409017537553063772923393142099200
1358886206848035256499499529324705317616144509789698561465481947644010269028371888510877653986718213835685764625880354818173798776095501719176632383682072619960298764422400
1490944659008005641792166016015573600453902425406350581830427401346328356020679277569350411257324739961344836558808184130859900761592768029934670412987938363412036836122466417681669615661129838688115200
Technical details
Node id
n-e146faf479fcfad1 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-72c8cf9569431b31

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        3;
  2:       15,        18;
  3:       78,       162,        81;
  4:      462,      1278,      1134,       324;
  5:     3132,     10440,     12555,      6480,      1215;
  6:    24084,     91872,    134865,     95580,     32805,     4374;
  7:   207792,    879480,   1492344,   1304100,    620865,   153090,    15309;
  8:  1991088,   9162072,  17388756,  17706276,  10478160,  3612924,   673596,   52488;
  9: 20999520, 103567248, 215116884, 246617784, 171180135, 73850616, 19381194, 2834352, 177147;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+6*n*x+9*x^2)*P(n-2,x) + (2*n+1+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n*x + 9*x^2)*P[n - 2, x] + (2*n + 1 + 6*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] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n*x + 9*x^2)*v[i - 2] + (2*n + 1 + 6*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. [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.