Real-rooted atlas

A134991 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
233
3103040
425210315550
556980378037808596
61193836283506930051975153580
72461365017094076230013513508108103109296
850145870908985654654020495475283783501418917570564896

All 15 rows

Technical details
Node id
n-a9776fec91eee7a3 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-064a056e3e3cfaf4

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:    3;
   3:   10,     30;
   4:   25,    210,      315;
   5:   56,    980,     3780,      3780;
   6:  119,   3836,    28350,     69300,      51975;
   7:  246,  13650,   170940,    762300,    1351350,     810810;
   8:  501,  45870,   908985,   6546540,   20495475,   28378350,    14189175;
   9: 1012, 148632,  4461600,  48488440,  237537300,  567567000,   643242600,   275675400;
  10: 2035, 470184, 20744724, 325725400, 2334382050, 8533685160, 16402686300, 15713497800, 5892561675;
%F Formula
The row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2+4*n+2)*x+(n^3+2*n^2+n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2+2*n+1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+(2*n^2+4*n)*x)*P(n-1,x) + (n*x+n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 + 4*n + 2)*x + (n^3 + 2*n^2 + n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 + 2*n + 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + (2*n^2 + 4*n)*x)*P[n - 1, x] + (n*x + n*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(2*n + 2 + (2*n^2 + 4*n + 2)*x + (n^3 + 2*n^2 + n)*x^2)*v[i - 2] - ((n + 1)*x + (n^2 + 2*n + 1)*x^2 + (n^2 + n)*x^3)*deriv(v[i - 2], x) + (3*n + 1 + (2*n^2 + 4*n)*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, 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.