Real-rooted atlas

A008292 → γ (scale 1/4) → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3Pn(1)
111
222
38412
4403272
524027248560
6168024648164960
7134402419211904108850624
812096025804816896031744579712

All 15 rows

Technical details
Node id
n-62e9e162e1ea8a56 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-44df5437d7babb90

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:        2;
   3:        8,        4;
   4:       40,       32;
   5:      240,      272,       48;
   6:     1680,     2464,      816;
   7:    13440,    24192,    11904,     1088;
   8:   120960,   258048,   168960,    31744;
   9:  1209600,  2983680,  2442240,   707584,   39680;
  10: 13305600, 37255680, 36622080, 14440448, 1768960;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-3*n+(2*n^2-6*n)*x+(n^2-3*n)*x^2)*P(n-2,x) + (2*n*x+4*n*x^2+2*n*x^3)*d/dx P(n-2,x) + (2*n-2+(2*n-4)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 3*n + (2*n^2 - 6*n)*x + (n^2 - 3*n)*x^2)*P[n - 2, x] + (2*n*x + 4*n*x^2 + 2*n*x^3)*D[P[n - 2, x], x] + (2*n - 2 + (2*n - 4)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = -(n^2 - 3*n + (2*n^2 - 6*n)*x + (n^2 - 3*n)*x^2)*v[i - 2] + (2*n*x + 4*n*x^2 + 2*n*x^3)*deriv(v[i - 2], x) + (2*n - 2 + (2*n - 4)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], 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

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.