Real-rooted atlas

A075504 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 18\,x + 9\,n x\right)\,P_{n-1}(x) + \left(-9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 27\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
012781109
11545677291351
2190202510935656119612
3113552656561020339159049333451
4118911340255150197486137200875314416493069
51252215467654501116026157041334674930074782969141264820
613243742219289344560551145881073016076090251219657095430467213376695763
71405607504286520149767947251105872518124795305446251007702199687443138742048987799365343

All 15 rows

Technical details
Node id
n-a1d96d78bb1f4642 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f08b49fd82dd971b

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1,  27,     81;
  1: 1,  54,    567,      729;
  2: 1,  90,   2025,    10935,       6561;
  3: 1, 135,   5265,    65610,     203391,       59049;
  4: 1, 189,  11340,   255150,    1974861,     3720087,       531441;
  5: 1, 252,  21546,   765450,   11160261,    57041334,     67493007,       4782969;
  6: 1, 324,  37422,  1928934,   45605511,   458810730,   1607609025,    1219657095,       43046721;
  7: 1, 405,  60750,  4286520,  149767947,  2511058725,  18124795305,   44625100770,    21996874431,      387420489;
  8: 1, 495,  93555,  8660520,  419818707, 10598527863, 131122437930,  697117731750,  1226874595221,   396331160247,    3486784401;
  9: 1, 594, 138105, 16238475, 1043376147, 37047106404, 703442942532, 6597627438600, 26323112938221, 33521945231214, 7137447668847, 31381059609;
%F Formula
The row polynomials satisfy P(n,x) = (1+(9*n+18)*x)*P(n-1,x) - 9*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 27*x + 81*x^2.
%t Mathematica
P[0, x_] := 1 + 27*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (9*n + 18)*x)*P[n - 1, x] - 9*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 27*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (9*n + 18)*x)*v[i - 1] - 9*x^2*deriv(v[i - 1], x)); 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.