Real-rooted atlas

A008277 → θ+1 → D+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
3312722
4430501599
556019518031471
661055601050602632386
77168133042005103193212712867
882522772132302780423310605025573681

All 15 rows

Technical details
Node id
n-1b6003327f97748a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-90c2a235bbf6b62e

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  2,   3;
   3:  3,  12,    7;
   4:  4,  30,   50,    15;
   5:  5,  60,  195,   180,     31;
   6:  6, 105,  560,  1050,    602,     63;
   7:  7, 168, 1330,  4200,   5103,   1932,    127;
   8:  8, 252, 2772, 13230,  27804,  23310,   6050,    255;
   9:  9, 360, 5250, 35280, 114135, 170100, 102315,  18660,   511;
  10: 10, 495, 9240, 83160, 383922, 897435, 986920, 437250, 57002, 1023;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(n-1)*x+(n-1)*x^2)*P(n-2,x) + (x^2+x^3)*d/dx P(n-2,x) + (2+(n+1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(1 + (n - 1)*x + (n - 1)*x^2)*P[n - 2, x] + (x^2 + x^3)*D[P[n - 2, x], x] + (2 + (n + 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 3*x; for(n = 3, N, my(i = n); v[i] = -(1 + (n - 1)*x + (n - 1)*x^2)*v[i - 2] + (x^2 + x^3)*deriv(v[i - 2], x) + (2 + (n + 1)*x)*v[i - 1] - x^2*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.