Real-rooted atlas

A173018 → reverse (shift +1) → θ+1 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
122
2325
3412218
454433284
56130264782480
673421510120817123240
788407146120804764360225200

All 15 rows

Technical details
Node id
n-b8f58c786d069403 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-134496fa7b090659

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  2:  3,    2;
  3:  4,   12,      2;
  4:  5,   44,     33,      2;
  5:  6,  130,    264,     78,      2;
  6:  7,  342,   1510,   1208,    171,      2;
  7:  8,  840,   7146,  12080,   4764,    360,     2;
  8:  9, 1976,  30051,  93714,  78095,  17172,   741,    2;
  9: 10, 4518, 116864, 617638, 937140, 441170, 58432, 1506, 2;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^2+n+(n^3-2*n^2-3*n)*x-(n^2-n-2)*x^2)*P(n-2,x) - ((n^2+n)*x-(n^2+2*n+1)*x^2+(n+1)*x^3)*d/dx P(n-2,x) + (n^2+n+(n^2-2*n-1)*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n + (n^3 - 2*n^2 - 3*n)*x - (n^2 - n - 2)*x^2)*P[n - 2, x] - ((n^2 + n)*x - (n^2 + 2*n + 1)*x^2 + (n + 1)*x^3)*D[P[n - 2, x], x] + (n^2 + n + (n^2 - 2*n - 1)*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + n + (n^3 - 2*n^2 - 3*n)*x - (n^2 - n - 2)*x^2)*v[i - 2] - ((n^2 + n)*x - (n^2 + 2*n + 1)*x^2 + (n + 1)*x^3)*deriv(v[i - 2], x) + (n^2 + n + (n^2 - 2*n - 1)*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); 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

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.