Real-rooted atlas

A079621 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 - x + n\right)\,P_{n}(x) = \left(8 - x^{2} + 10\,n - n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(4\,x - 2\,x^{2} + 2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1516
23614151
3336192271556
4384028806004417365
55184048000132001440651114546
68064008870403024004368029409012042551
71419264018063360733824013171201176005376119141034456

All 15 rows

Technical details
Node id
n-573b9b78f7c813d9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-48f9e57f7812cd06

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:          5,          1;
  2:         36,         14,          1;
  3:        336,        192,         27,          1;
  4:       3840,       2880,        600,         44,         1;
  5:      51840,      48000,      13200,       1440,        65,        1;
  6:     806400,     887040,     302400,      43680,      2940,       90,      1;
  7:   14192640,   18063360,    7338240,    1317120,    117600,     5376,    119,     1;
  8:  278691840,  402554880,  189665280,   40642560,   4515840,   274176,   9072,   152,   1;
  9: 6038323200, 9754214400, 5225472000, 1300561920, 172730880, 13063680, 574560, 14400, 189, 1;
%F Formula
The row polynomials satisfy (n+3-x)*P(n,x) = (2*n^2+10*n+8-n*x-x^2)*P(n-1,x) + ((2*n+4)*x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 10*n + 8 - n*x - x^2)*P[n - 1, x] + ((2*n + 4)*x - 2*x^2)*D[P[n - 1, x], x])/(n + 3 - x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 10*n + 8 - n*x - x^2)*v[i - 1] + ((2*n + 4)*x - 2*x^2)*deriv(v[i - 1], x))/(n + 3 - 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  4. [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
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  7. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023