Real-rooted atlas

A330798 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(4 + 3\,x - 20\,n - 12\,n x + 8\,n^{2} + 3\,n^{2} x + 32\,n^{3} + 18\,n^{3} x\right)\,P_{n-1}(x) + \left(-8 - 10\,x - 3\,x^{2} + 12\,n + 18\,n x + 6\,n x^{2} + 20\,n^{2} + 28\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
23033972
3264452256481020
4249057955025192527515510
524396718148409648984141961638245124
62447908716051286985100891044296010329299963968538
724965281043021618597888183491201082016038139847441926201665314104

All 15 rows

Technical details
Node id
n-6fe98fbfbcdc3b9a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a09e6eb68394f014

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:         4,          2;
  2:        30,         33,          9;
  3:       264,        452,        256,         48;
  4:      2490,       5795,       5025,       1925,        275;
  5:     24396,      71814,      84096,      48984,      14196,       1638;
  6:    244790,     871605,    1286985,    1008910,     442960,     103292,       9996;
  7:   2496528,   10430216,   18597888,   18349120,   10820160,    3813984,     744192,     62016;
  8:  25763058,  123542919,  258229377,  307321371,  227795085,  107697681,   31719303,   5321349,   389367;
  9: 268243860, 1451989130, 3481484800, 4853668400, 4336255000, 2574715220, 1016128960, 257048000, 37823500, 2466750;
%F Formula
The row polynomials satisfy (4*n^3+2*n^2)*P(n,x) = (32*n^3+8*n^2-20*n+4+(18*n^3+3*n^2-12*n+3)*x)*P(n-1,x) + (20*n^2+12*n-8+(28*n^2+18*n-10)*x+(9*n^2+6*n-3)*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[Together[((32*n^3 + 8*n^2 - 20*n + 4 + (18*n^3 + 3*n^2 - 12*n + 3)*x)*P[n - 1, x] + (20*n^2 + 12*n - 8 + (28*n^2 + 18*n - 10)*x + (9*n^2 + 6*n - 3)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 2*n^2)]];
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] = ((32*n^3 + 8*n^2 - 20*n + 4 + (18*n^3 + 3*n^2 - 12*n + 3)*x)*v[i - 1] + (20*n^2 + 12*n - 8 + (28*n^2 + 18*n - 10)*x + (9*n^2 + 6*n - 3)*x^2)*deriv(v[i - 1], x))/(4*n^3 + 2*n^2)); 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. [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.