Real-rooted atlas

A156289 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-1}(x) + \left(5 + 7\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2437
331451591
43797985251051807
565561789517325693094549651
6150349500643634425377685103950103951777447
743734611710618026636610209309108648640175675513513579587691
8156297964699786363128718135012509897406871614752109457353310807520270254327497727

All 15 rows

Technical details
Node id
n-992ded70fabcb6bc · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-27428437fc285c33

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:            1;
   2:            4,             3;
   3:           31,            45,            15;
   4:          379,           798,           525,           105;
   5:         6556,         17895,         17325,          6930,           945;
   6:       150349,        500643,        634425,        377685,        103950,         10395;
   7:      4373461,      17106180,      26636610,      20930910,       8648640,       1756755,       135135;
   8:    156297964,     699786363,    1287181350,    1250989740,     687161475,     210945735,     33108075,      2027025;
   9:   6698486371,   33693021225,   71265219495,   82244970990,   56198727585,   23069436390,   5499724230,    689188500,    34459425;
  10: 337789490599, 1881901591458, 4488768443775, 5982583378695, 4877453844900, 2503203848145, 801824873850, 153206603550, 15713497800, 654729075;
%F Formula
The row polynomials satisfy P(n,x) = (4+3*x)*P(n-1,x) + (5+7*x+2*x^2)*d/dx P(n-1,x) + (1+2*x+x^2)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + 3*x)*P[n - 1, x] + (5 + 7*x + 2*x^2)*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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (4 + 3*x)*v[i - 1] + (5 + 7*x + 2*x^2)*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, v[i])))};
T(10)

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.