Real-rooted atlas

A145596 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-4 + 3\,n + n^{2}\right)\,P_{n}(x) = \left(6 - 2\,x + 10\,n + 4\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
214620
3726412148
433045018020980
5143026401650400306150
66006140141201246207704237464
72475269888764404076810920134456224168

All 15 rows

Technical details
Node id
n-e9fa4da31855e6d0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cc9f40497aa70b3e

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      2;
  2:     14,       6;
  3:     72,      64,      12;
  4:    330,     450,     180,      20;
  5:   1430,    2640,    1650,     400,     30;
  6:   6006,   14014,   12012,    4620,    770,     42;
  7:  24752,   69888,   76440,   40768,  10920,   1344,    56;
  8: 100776,  334152,  445536,  305760, 114660,  22932,  2184,   72;
  9: 406980, 1550400, 2441880, 2056320, 999600, 282240, 44100, 3360, 90;
%F Formula
The row polynomials satisfy (n^2+3*n-4)*P(n,x) = (4*n^2+10*n+6+(3*n^2+4*n-2)*x)*P(n-1,x) - ((2*n+3)*x+(2*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 10*n + 6 + (3*n^2 + 4*n - 2)*x)*P[n - 1, x] - ((2*n + 3)*x + (2*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 + 3*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 10*n + 6 + (3*n^2 + 4*n - 2)*x)*v[i - 1] - ((2*n + 3)*x + (2*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 + 3*n - 4)); 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. [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.