Real-rooted atlas

A225470 → basis A008277 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 + 6\,x - x^{2} + 15\,n - 6\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(3\,x - x^{2} - 3\,n x\right)\,P_{n-2}'(x) + \left(-2 + 2\,x + 6\,n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
210212
3118423163
4169682410842632
529080174003180220549885
658168040616092880908039061090196
71332184010505040283836035084021490630727038207

All 15 rows

Technical details
Node id
n-8b8ef2048fe11985 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d909fb09b4b82f96

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:         10,          2;
  3:        118,         42,          3;
  4:       1696,        824,        108,         4;
  5:      29080,      17400,       3180,       220,        5;
  6:     581680,     406160,      92880,      9080,      390,       6;
  7:   13321840,   10505040,    2838360,    350840,    21490,     630,     7;
  8:  344068480,  299774720,   92393280,  13608000,  1061760,   44688,   952,    8;
  9: 9898456960, 9381603840, 3221460480, 545039040, 50986320, 2748816, 84504, 1368, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-15*n+6+(6*n-6)*x+x^2)*P(n-2,x) - ((3*n-3)*x+x^2)*d/dx P(n-2,x) + (6*n-2+2*x)*P(n-1,x) + x*d/dx 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[-(9*n^2 - 15*n + 6 + (6*n - 6)*x + x^2)*P[n - 2, x] - ((3*n - 3)*x + x^2)*D[P[n - 2, x], x] + (6*n - 2 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
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] = -(9*n^2 - 15*n + 6 + (6*n - 6)*x + x^2)*v[i - 2] - ((3*n - 3)*x + x^2)*deriv(v[i - 2], x) + (6*n - 2 + 2*x)*v[i - 1] + x*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 - 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. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.