Real-rooted atlas

A166962 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 2\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(4 + x - 3\,x^{2}\right)\,P_{n-1}'(x) + \left(3 + 6\,x + 3\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21414129
3174244721491
4332261263124320112893
5897562033161453843315613341472947
632114208600272791722028312203082705418122873821
71461328084509290525062827922464567204756701026186242178811400008795

All 15 rows

Technical details
Node id
n-2feacf100ed042e2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4364dfc5bf2ac04c

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:           14,            14,             1;
  3:          174,           244,            72,             1;
  4:         3322,          6126,          3124,           320,            1;
  5:        89756,        203316,        145384,         33156,         1334,            1;
  6:      3211420,       8600272,       7917220,       2831220,       308270,         5418,          1;
  7:    146132808,     450929052,     506282792,     246456720,     47567010,      2618624,      21788,         1;
  8:   8202467544,   28599095824,   37804753460,   23287634672,   6569227270,    710553890,   20923420,     87308,      1;
  9: 554489060200, 2152509769144, 3268840371744, 2433534378484, 907531806972, 154339736988, 9682642548, 160346280, 349434, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-1+(3*n-2)*x)*P(n-1,x) + (4+x-3*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[(3*n - 1 + (3*n - 2)*x)*P[n - 1, x] + (4 + x - 3*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] = (3*n - 1 + (3*n - 2)*x)*v[i - 1] + (4 + x - 3*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.