Real-rooted atlas

A114655 → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
122
28412
33836680
4192264968560
5100218001020200104032
6533611820912029203601229568
728814758527413033320693058814219648
81571844792485671683298409856014448896161647360

All 15 rows

Technical details
Node id
n-a1c963de13c50910 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bf35de5d7524cff0

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:       2;
   2:       8,        4;
   3:      38,       36,        6;
   4:     192,      264,       96,        8;
   5:    1002,     1800,     1020,      200,       10;
   6:    5336,    11820,     9120,     2920,      360,      12;
   7:   28814,    75852,    74130,    33320,     6930,     588,     14;
   8:  157184,   479248,   567168,   329840,    98560,   14448,    896,    16;
   9:  864146,  2994192,  4161528,  2976624,  1170540,  250992,  27384,  1296,   18;
  10: 4780008, 18548820, 29612160, 25171440, 12332880, 3528504, 571200, 48240, 1800, 20;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (5*n^2-7*n+2+(3*n^2-5*n+2)*x)*P(n-1,x) + (2*n-1-(2*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 7*n + 2 + (3*n^2 - 5*n + 2)*x)*P[n - 1, x] + (2*n - 1 - (2*n - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((5*n^2 - 7*n + 2 + (3*n^2 - 5*n + 2)*x)*v[i - 1] + (2*n - 1 - (2*n - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.