Real-rooted atlas

A371081 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + x + 6\,n + n^{2}\right)\,P_{n-1}(x) + \left(7 + 7\,x + 2\,n + 2\,n x\right)\,P_{n-1}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
117118
2453541508
3174653183119120768
49212332284121299822011162864
56378914520146261155508239930365185530784
656165990132164114530212625815738145210211556218000823488
7613148157073279463647875335496919851478512315275796272293138191927667819008

All 15 rows

Technical details
Node id
n-09268dddee3be1b7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-78aed4e5d7efa0b0

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:                17,                1;
  2:               453,               54,                1;
  3:             17465,             3183,              119,              1;
  4:            921233,           228412,            12998,            220,             1;
  5:          63789145,         20146261,          1555082,          39930,           365,           1;
  6:        5616599013,       2164114530,        212625815,        7381452,        102115,         562,         1;
  7:      613148157073,     279463647875,      33549691985,     1478512315,      27579627,      229313,       819,      1;
  8:    81298838448001,   42821732772088,    6103001738364,   328020441032,    7747076102,    86675272,    467068,   1144,    1;
  9: 12871080897739073, 7692523626220809, 1273743818512804, 81197992353684, 2333334431166, 32981037310, 238768724, 881508, 1545, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^2+6*n+10+x)*P(n-1,x) + (2*n+7+(2*n+7)*x)*d/dx P(n-1,x) + (1+2*x+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^2 + 6*n + 10 + x)*P[n - 1, x] + (2*n + 7 + (2*n + 7)*x)*D[P[n - 1, x], x] + (1 + 2*x + 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^2 + 6*n + 10 + x)*v[i - 1] + (2*n + 7 + (2*n + 7)*x)*deriv(v[i - 1], x) + (1 + 2*x + 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.