Real-rooted atlas

A390433 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-36 - 18\,x - x^{2} + 60\,n + 12\,n x - 37\,n^{2} - 2\,n^{2} x + 10\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(27\,x + 4\,x^{2} - 36\,n x - 2\,n x^{2} + 15\,n^{2} x - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-9\,x^{2} - x^{3} + 6\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(10 + 2\,x - 8\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(-x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3426
43640379
5576107216841820
614400397449840480564469
7518400197625668937652000110063237138
82540160012751718458803840611916819810021847218042083

All 15 rows

Technical details
Node id
n-9e89907ee1f3ed20 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-bbde493405ec4710

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 -2)
  • 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:            0;
   2:            1;
   3:            4,             2;
   4:           36,            40,            3;
   5:          576,          1072,          168,            4;
   6:        14400,         39744,         9840,          480,          5;
   7:       518400,       1976256,       689376,        52000,       1100,         6;
   8:     25401600,     127517184,     58803840,      6119168,     198100,      2184,       7;
   9:   1625702400,   10379695104,   6071659776,    818824448,   36175360,    606816,    3920,    8;
  10: 131681894400, 1041220915200, 750240375552, 126006266880, 7137166400, 162346368, 1589952, 6528, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-10*n^3+37*n^2-60*n+36+(2*n^2-12*n+18)*x+x^2)*P(n-2,x) - ((2*n^3-15*n^2+36*n-27)*x+(2*n-4)*x^2)*d/dx P(n-2,x) - ((n^2-6*n+9)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2-8*n+10+2*x)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 10*n^3 + 37*n^2 - 60*n + 36 + (2*n^2 - 12*n + 18)*x + x^2)*P[n - 2, x] - ((2*n^3 - 15*n^2 + 36*n - 27)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] - ((n^2 - 6*n + 9)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 8*n + 10 + 2*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(n^4 - 10*n^3 + 37*n^2 - 60*n + 36 + (2*n^2 - 12*n + 18)*x + x^2)*v[i - 2] - ((2*n^3 - 15*n^2 + 36*n - 27)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) - ((n^2 - 6*n + 9)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 8*n + 10 + 2*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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

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.