Real-rooted atlas

A243662 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(10 + 4\,x - 43\,n - 16\,n x + 39\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(5 + 7\,x + 2\,x^{2} - 17\,n - 25\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2415
32110132
412688181233
58187522342811833
655946375267050040115180
73969353966284437300935541130392
8289510457240290640950601694015967011151057

All 15 rows

Technical details
Node id
n-3a57a47e218ae50e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-84e6bc5890e6c39e

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        4,        1;
   3:       21,       10,        1;
   4:      126,       88,       18,        1;
   5:      818,      752,      234,       28,       1;
   6:     5594,     6375,     2670,      500,      40,      1;
   7:    39693,    53966,    28443,     7300,     935,     54,     1;
   8:   289510,   457240,   290640,    95060,   16940,   1596,    70,    1;
   9:  2157150,  3880720,  2889040,  1150240,  264040,  35056,  2548,   88,   1;
  10: 16348960, 33001362, 28164744, 13227408, 3726156, 643230, 66612, 3864, 108, 1;
%F Formula
The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (39*n^2-43*n+10+(12*n^2-16*n+4)*x)*P(n-1,x) - (17*n-5+(25*n-7)*x+(8*n-2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((39*n^2 - 43*n + 10 + (12*n^2 - 16*n + 4)*x)*P[n - 1, x] - (17*n - 5 + (25*n - 7)*x + (8*n - 2)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((39*n^2 - 43*n + 10 + (12*n^2 - 16*n + 4)*x)*v[i - 1] - (17*n - 5 + (25*n - 7)*x + (8*n - 2)*x^2)*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, v[i])))};
T(10)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.