Real-rooted atlas

A166960 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-4\,n + 4\,n x - 2\,n x^{2} - 2\,n^{2} x + 3\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-5\,n x + 5\,n x^{2} - n x^{3} - 4\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-n x^{2} + n x^{3} - n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-4 + 2\,x + 5\,n - 4\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-5\,x + x^{2} + 5\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(-x^{2} + n x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2628
32742372
41125401744830
545357565214564511992
61818563341179603495615906212664
7727952882223098651503020188325411074541428

All 15 rows

Technical details
Node id
n-c1338451c8a75e40 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-15d50c9a2b2cc2b4

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:      1;
  2:      6,        2;
  3:     27,       42,         3;
  4:    112,      540,       174,          4;
  5:    453,     5756,      5214,        564,         5;
  6:   1818,    56334,    117960,      34956,      1590,         6;
  7:   7279,   528822,   2309865,    1503020,    188325,      4110,       7;
  8:  29124,  4861304,  41717238,   52974600,  14294800,    879360,   10038,     8;
  9: 116505, 44217720, 718519500, 1658102904, 845704050, 111703680, 3720192, 23592, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (5*n*x+(4*n^2-5*n)*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) - (n*x^2+(n^2-n)*x^3)*d^2/dx^2 P(n-2,x) + (5*n-4+(2*n^2-4*n+2)*x)*P(n-1,x) + ((5*n-5)*x-(n-1)*x^2)*d/dx P(n-1,x) + (n-1)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (5*n*x + (4*n^2 - 5*n)*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] - (n*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], {x, 2}] + (5*n - 4 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((5*n - 5)*x - (n - 1)*x^2)*D[P[n - 1, x], x] + (n - 1)*x^2*D[P[n - 1, x], {x, 2}])/(n - 1)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (5*n*x + (4*n^2 - 5*n)*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) - (n*x^2 + (n^2 - n)*x^3)*deriv(deriv(v[i - 2], x), x) + (5*n - 4 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((5*n - 5)*x - (n - 1)*x^2)*deriv(v[i - 1], x) + (n - 1)*x^2*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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. [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.