Real-rooted atlas

A166972 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(1 + n + n x\right)\,P_{n-1}(x) + \left(6 + 5\,x - x^{2}\right)\,P_{n-1}'(x) + \left(3 + 6\,x + 3\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21212125
3126168441339
419023258148212616769
539852831845247694563141185283
61092924266695221447966183484830072016572041
73761388010492621210260640840756248567268221207615681291789075

All 15 rows

Technical details
Node id
n-95aa1bab8730e108 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4f720f55b3b0bb67

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           2,            1;
  2:          12,           12,            1;
  3:         126,          168,           44,            1;
  4:        1902,         3258,         1482,          126,           1;
  5:       39852,        83184,        52476,         9456,         314,          1;
  6:     1092924,      2666952,      2144796,       618348,       48300,        720,         1;
  7:    37613880,    104926212,    102606408,     40756248,     5672682,     212076,      1568,       1;
  8:  1583720640,   4966044264,   5704697628,   2884702896,   604458090,   42964686,    838794,    3306,    1;
  9: 79861657752, 277581544920, 365584940064, 224580965724, 63722645676, 7285531680, 281874096, 3082092, 6830, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+1+n*x)*P(n-1,x) + (6+5*x-x^2)*d/dx P(n-1,x) + (3+6*x+3*x^2)*d^2/dx^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[(n + 1 + n*x)*P[n - 1, x] + (6 + 5*x - x^2)*D[P[n - 1, x], x] + (3 + 6*x + 3*x^2)*D[P[n - 1, x], {x, 2}]];
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] = (n + 1 + n*x)*v[i - 1] + (6 + 5*x - x^2)*deriv(v[i - 1], x) + (3 + 6*x + 3*x^2)*deriv(deriv(v[i - 1], x), 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

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.