Real-rooted atlas

A112486 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
25611
3267045141
41546809454202199
5104466081470013860472540937
680286748021159631500022522562370889699
769264735768302694062862807033950405405094594522152197

All 15 rows

Technical details
Node id
n-fd5f222f6785be4b · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-49ab8a1f300b12f8

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:       5,         6;
  3:      26,        70,        45;
  4:     154,       680,       945,        420;
  5:    1044,      6608,     14700,      13860,       4725;
  6:    8028,     67480,    211596,     315000,     225225,      62370;
  7:   69264,    735768,   3026940,    6286280,    7033950,    4054050,     945945;
  8:  663696,   8604432,  44332860,  119830480,  185735550,  166486320,   80405325,   16216200;
  9: 6999840, 107922672, 673967448, 2267122000, 4547543000, 5617652040, 4199995800, 1745944200, 310134825;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n^2*x+n^2*x^2)*P(n-2,x) - (n*x+2*n*x^2+n*x^3)*d/dx P(n-2,x) + (2*n+1+(2*n+2)*x)*P(n-1,x) + (x+x^2)*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[-(n^2 + 2*n^2*x + n^2*x^2)*P[n - 2, x] - (n*x + 2*n*x^2 + n*x^3)*D[P[n - 2, x], x] + (2*n + 1 + (2*n + 2)*x)*P[n - 1, x] + (x + x^2)*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] = -(n^2 + 2*n^2*x + n^2*x^2)*v[i - 2] - (n*x + 2*n*x^2 + n*x^3)*deriv(v[i - 2], x) + (2*n + 1 + (2*n + 2)*x)*v[i - 1] + (x + 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 - 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. [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.