Real-rooted atlas

A085880 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n x + n^{2}\right)\,P_{n}(x) = \left(-12 - 18\,x - 6\,x^{2} - 4\,n + 6\,n x + 4\,n x^{2} + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
31612230
4100120455270
56721120672168142646
6470410080840033606304227216
733792887049504052800158402376132288684
824710476876810090087207203003007207290094293127410

All 15 rows

Technical details
Node id
n-12985dc6a610374e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        3,        1;
   3:       16,       12,         2;
   4:      100,      120,        45,         5;
   5:      672,     1120,       672,       168,       14;
   6:     4704,    10080,      8400,      3360,      630,       42;
   7:    33792,    88704,     95040,     52800,    15840,     2376,     132;
   8:   247104,   768768,   1009008,    720720,   300300,    72072,    9009,     429;
   9:  1830400,  6589440,  10250240,   8968960,  4804800,  1601600,  320320,   34320,   1430;
  10: 13691392, 56010240, 100818432, 104552448, 68612544, 29405376, 8168160, 1400256, 131274, 4862;
%F Formula
The row polynomials satisfy (n^2+n*x)*P(n,x) = (8*n^2-4*n-12+(4*n^2+6*n-18)*x+(4*n-6)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 4*n - 12 + (4*n^2 + 6*n - 18)*x + (4*n - 6)*x^2)*P[n - 1, x])/(n^2 + n*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^2 - 4*n - 12 + (4*n^2 + 6*n - 18)*x + (4*n - 6)*x^2)*v[i - 1])/(n^2 + n*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.