Real-rooted atlas

A292219 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 10\,x + 48\,x^{2} + 8\,n x - 56\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x^{2} + 72\,x^{3} - 32\,n x^{3}\right)\,P_{n-1}'(x) + \left(16\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2167
31206081
41424208401303
51721512100801512026785
61110396055440277200332640669351
71156858020592021621608648640864864019674097
81210163806006001081080090810720302702400259459200664400311

All 15 rows

Technical details
Node id
n-1658117d1dedf548 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f136838b00e1c323

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: 1,   6;
   3: 1,  20,    60;
   4: 1,  42,   420,     840;
   5: 1,  72,  1512,   10080,     15120;
   6: 1, 110,  3960,   55440,    277200,     332640;
   7: 1, 156,  8580,  205920,   2162160,    8648640,     8648640;
   8: 1, 210, 16380,  600600,  10810800,   90810720,   302702400,    259459200;
   9: 1, 272, 28560, 1485120,  40840800,  588107520,  4116752640,  11762150400,   8821612800;
  10: 1, 342, 46512, 3255840, 126977760, 2793510720, 33522128640, 201132771840, 502831929600, 335221286400;
%F Formula
The row polynomials satisfy P(n,x) = (1+(8*n-10)*x+(16*n^2-56*n+48)*x^2)*P(n-1,x) - (8*x^2+(32*n-72)*x^3)*d/dx P(n-1,x) + 16*x^4*d^2/dx^2 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[(1 + (8*n - 10)*x + (16*n^2 - 56*n + 48)*x^2)*P[n - 1, x] - (8*x^2 + (32*n - 72)*x^3)*D[P[n - 1, x], x] + 16*x^4*D[P[n - 1, x], {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] = (1 + (8*n - 10)*x + (16*n^2 - 56*n + 48)*x^2)*v[i - 1] - (8*x^2 + (32*n - 72)*x^3)*deriv(v[i - 1], x) + 16*x^4*deriv(deriv(v[i - 1], x), x)); 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.