Real-rooted atlas

A136124 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2527
32618347
4154142424342
5104411604658052754
680281020849951180135624552
76926497720552721610025552107241128

All 15 rows

Technical details
Node id
n-02b8fac5a8097d07 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6443481be5042dac

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,        2;
  3:      26,       18,       3;
  4:     154,      142,      42,       4;
  5:    1044,     1160,     465,      80,      5;
  6:    8028,    10208,    4995,    1180,    135,      6;
  7:   69264,    97720,   55272,   16100,   2555,    210,    7;
  8:  663696,  1018008,  644028,  218596,  43120,   4956,  308,   8;
  9: 6999840, 11507472, 7967292, 3044664, 704445, 101304, 8862, 432, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n*x+x^2)*P(n-2,x) + (2*n+1+2*x)*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*x + x^2)*P[n - 2, x] + (2*n + 1 + 2*x)*P[n - 1, 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*x + x^2)*v[i - 2] + (2*n + 1 + 2*x)*v[i - 1]); 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

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.