Real-rooted atlas

A243662 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 6\,n x + 2\,n x^{2} - 4\,n^{2} - 8\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{3} + 8\,n^{3} x + 4\,n^{4}\right)\,P_{n}(x) = \left(-12 - 26\,x - 2\,x^{2} + 4\,x^{3} + 60\,n + 121\,n x + 13\,n x^{2} - 16\,n x^{3} - 75\,n^{2} - 159\,n^{2} x - 29\,n^{2} x^{2} + 12\,n^{2} x^{3} + 46\,n^{3} x + 24\,n^{3} x^{2} + 27\,n^{4} + 12\,n^{4} x\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} + x^{3} + 2\,x^{4} + 16\,n x + 32\,n x^{2} + 3\,n x^{3} - 8\,n x^{4} - 9\,n^{2} x - 28\,n^{2} x^{2} - 16\,n^{2} x^{3} - 9\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2415
32010131
411085181214
56376762282811570
638085180245049040111969
7232563876023868680092054193659
814421028528521785481795159601575701746750

All 15 rows

Technical details
Node id
n-0d114e4d0151ef21 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1d7ecb93c762a6b7

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:      20,       10,        1;
   4:     110,       85,       18,       1;
   5:     637,      676,      228,      28,       1;
   6:    3808,     5180,     2450,     490,      40,      1;
   7:   23256,    38760,    23868,    6800,     920,     54,     1;
   8:  144210,   285285,   217854,   81795,   15960,   1575,    70,    1;
   9:  904475,  2074600,  1898512,  892584,  231770,  33320,  2520,   88,   1;
  10: 5722860, 14948505, 15984540, 9077640, 2975280, 573804, 63756, 3828, 108, 1;
%F Formula
The row polynomials satisfy (4*n^4+2*n^3-4*n^2-2*n+(8*n^3-8*n^2-6*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (27*n^4-75*n^2+60*n-12+(12*n^4+46*n^3-159*n^2+121*n-26)*x+(24*n^3-29*n^2+13*n-2)*x^2+(12*n^2-16*n+4)*x^3)*P(n-1,x) - ((9*n^3+9*n^2-16*n+4)*x+(8*n^3+28*n^2-32*n+8)*x^2+(16*n^2-3*n-1)*x^3+(8*n-2)*x^4)*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[((27*n^4 - 75*n^2 + 60*n - 12 + (12*n^4 + 46*n^3 - 159*n^2 + 121*n - 26)*x + (24*n^3 - 29*n^2 + 13*n - 2)*x^2 + (12*n^2 - 16*n + 4)*x^3)*P[n - 1, x] - ((9*n^3 + 9*n^2 - 16*n + 4)*x + (8*n^3 + 28*n^2 - 32*n + 8)*x^2 + (16*n^2 - 3*n - 1)*x^3 + (8*n - 2)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 2*n^3 - 4*n^2 - 2*n + (8*n^3 - 8*n^2 - 6*n)*x + (4*n^2 + 2*n)*x^2)]];
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] = ((27*n^4 - 75*n^2 + 60*n - 12 + (12*n^4 + 46*n^3 - 159*n^2 + 121*n - 26)*x + (24*n^3 - 29*n^2 + 13*n - 2)*x^2 + (12*n^2 - 16*n + 4)*x^3)*v[i - 1] - ((9*n^3 + 9*n^2 - 16*n + 4)*x + (8*n^3 + 28*n^2 - 32*n + 8)*x^2 + (16*n^2 - 3*n - 1)*x^3 + (8*n - 2)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 2*n^3 - 4*n^2 - 2*n + (8*n^3 - 8*n^2 - 6*n)*x + (4*n^2 + 2*n)*x^2)); 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

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.