Real-rooted atlas

A028421 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
236211
341822650
454010510024274
56753406755481201764
6712687529404872352872013068
78196193298002707639396261365040109584

All 15 rows

Technical details
Node id
n-00c8e241a6f9a311 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-aa53210b1406856d

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:  3,   6,    2;
  3:  4,  18,   22,     6;
  4:  5,  40,  105,   100,     24;
  5:  6,  75,  340,   675,    548,     120;
  6:  7, 126,  875,  2940,   4872,    3528,     720;
  7:  8, 196, 1932,  9800,  27076,   39396,   26136,    5040;
  8:  9, 288, 3822, 27216, 112245,  269136,  354372,  219168,   40320;
  9: 10, 405, 6960, 66150, 379638, 1346625, 2894720, 3518100, 2053152, 362880;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(2*n-2)*x+(n^2-2*n+1)*x^2)*P(n-2,x) + (2+(2*n-1)*x)*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[-(1 + (2*n - 2)*x + (n^2 - 2*n + 1)*x^2)*P[n - 2, x] + (2 + (2*n - 1)*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] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + (2*n - 2)*x + (n^2 - 2*n + 1)*x^2)*v[i - 2] + (2 + (2*n - 1)*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.