Real-rooted atlas

A088996 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
21815336
316222510215504
41944383427998941059576
5291607500676491386019621945229824
6524880167086822019041536219597789122841103956664896
71102248041911668679612686088535132524416103509181815228135135226606464

All 15 rows

Technical details
Node id
n-b5f3abea14d494ac · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-025882eec0fd4c19

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 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:          3,           1;
  2:         18,          15,           3;
  3:        162,         225,         102,           15;
  4:       1944,        3834,        2799,          894,         105;
  5:      29160,       75006,       76491,        38601,        9621,         945;
  6:     524880,     1670868,     2201904,      1536219,      597789,      122841,       10395;
  7:   11022480,    41911668,    67961268,     60885351,    32524416,    10350918,     1815228,     135135;
  8:  264539520,  1171217232,  2259745452,   2480667444,  1693866249,   736288272,   198829242,   30471660,   2027025;
  9: 7142567040, 36120037104, 80923820148, 105393693672, 87905735271, 48675509577, 17885290158, 4202831934, 572747895, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+(2*n-1)*x)*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[(3*n + (2*n - 1)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3*n + (2*n - 1)*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

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.