Real-rooted atlas

A258758 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(4 + 2\,x - 12\,n - 6\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 6\,n x + 4\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21215330
3801909010370
456022751925490355285
54032265863591015750252012684924
629568307230624162420420113190124744621507506
721964835332441041440410126116408408074474460060171629184012

All 15 rows

Technical details
Node id
n-ed6a65c768043ce7 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

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 0)
  • 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:        1;
  1:        2,         1;
  2:       12,        15,          3;
  3:       80,       190,         90,         10;
  4:      560,      2275,       1925,        490,         35;
  5:     4032,     26586,      35910,      15750,       2520,        126;
  6:    29568,    307230,     624162,     420420,     113190,      12474,       462;
  7:   219648,   3533244,   10414404,   10126116,    4084080,     744744,     60060,     1716;
  8:  1647360,  40572675,  169465725,  228918690,  129864735,   34864830,   4594590,   283140,    6435;
  9: 12446720, 466047010, 2714089950, 4964223550, 3808404600, 1412581170, 270570300, 26984100, 1312740, 24310;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (8*n^2-12*n+4+(4*n^2-6*n+2)*x)*P(n-1,x) + (4*n^2-6*n+2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 12*n + 4 + (4*n^2 - 6*n + 2)*x)*P[n - 1, x] + (4*n^2 - 6*n + 2)*x*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 12*n + 4 + (4*n^2 - 6*n + 2)*x)*v[i - 1] + (4*n^2 - 6*n + 2)*x*deriv(v[i - 1], x))/(n^2 - n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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.