Real-rooted atlas

A089231 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
23730673
3361444180241009
4436172204440120012017341
562701130830108300444009000720355951
610442052633442274743015162004662007560050408488117
71974817758475480737363765128536021226800522144070560040320230439553

All 15 rows

Technical details
Node id
n-8883972c91658a94 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8eec9eaea56e0be8

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,           2;
  2:         37,          30,           6;
  3:        361,         444,         180,          24;
  4:       4361,        7220,        4440,        1200,         120;
  5:      62701,      130830,      108300,       44400,        9000,         720;
  6:    1044205,     2633442,     2747430,     1516200,      466200,       75600,       5040;
  7:   19748177,    58475480,    73736376,    51285360,    21226800,     5221440,     705600,     40320;
  8:  417787921,  1421868744,  2105117280,  1769673024,   923136480,   305665920,   62657280,   7257600,   362880;
  9: 9770678101, 37600912890, 63984093480, 63153518400, 39817643040, 16616456640, 4584988800, 805593600, 81648000, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+1+2*n*x)*P(n-1,x) - (4+4*x+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[(4*n + 1 + 2*n*x)*P[n - 1, x] - (4 + 4*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n + 1 + 2*n*x)*v[i - 1] - (4 + 4*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. [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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.