Real-rooted atlas

A357613 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(12 + 6\,x - 54\,n - 28\,n x - n x^{2} + 52\,n^{2} + 26\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(6 + 9\,x + 3\,x^{2} - 20\,n - 32\,n x - 13\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
24132679
3377438165201000
43649563231927847013327
5363657001053010196803570252182887
6369305852000807510401720110385158409242557684
7380030510218670116338957259980267767558258269069343236245608

All 15 rows

Technical details
Node id
n-f721652c141580b8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-52349188d9f9d3f2

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:        41,         32,          6;
  3:       377,        438,        165,         20;
  4:      3649,       5632,       3192,        784,         70;
  5:     36365,      70010,      53010,      19680,       3570,       252;
  6:    369305,     852000,     807510,     401720,     110385,     15840,      924;
  7:   3800305,   10218670,   11633895,    7259980,    2677675,    582582,    69069,     3432;
  8:  39490049,  121266176,  161196000,  121024960,   56066920,  16388736,  2946944,   297440,   12870;
  9: 413442773, 1427499954, 2169942084, 1904517888, 1062646200, 390478032, 94379376, 14448096, 1268982, 48620;
%F Formula
The row polynomials satisfy (4*n^2-2*n)*P(n,x) = (52*n^2-54*n+12+(26*n^2-28*n+6)*x+(n^2-n)*x^2)*P(n-1,x) - (20*n-6+(32*n-9)*x+(13*n-3)*x^2+n*x^3)*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[((52*n^2 - 54*n + 12 + (26*n^2 - 28*n + 6)*x + (n^2 - n)*x^2)*P[n - 1, x] - (20*n - 6 + (32*n - 9)*x + (13*n - 3)*x^2 + n*x^3)*D[P[n - 1, x], x])/(4*n^2 - 2*n)]];
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] = ((52*n^2 - 54*n + 12 + (26*n^2 - 28*n + 6)*x + (n^2 - n)*x^2)*v[i - 1] - (20*n - 6 + (32*n - 9)*x + (13*n - 3)*x^2 + n*x^3)*deriv(v[i - 1], x))/(4*n^2 - 2*n)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.