Real-rooted atlas

A269944 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 - 20\,x + 9\,x^{2} + 40\,n + 16\,n x - 26\,n^{2} - 4\,n^{2} x + 8\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(20\,x^{2} + 4\,x^{3} - 16\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(7 + 4\,x - 6\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
2212923
310687216166
4100720873288252006
51700126401646164488753638160
644200335440456426196912328252196491068048
7163540012588080176425028097168152220012852048026441618736

All 15 rows

Technical details
Node id
n-d80751263aa1425b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e8f8aa9d3fc84e66

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:          1,           4;
  2:          2,          12,           9;
  3:         10,          68,          72,          16;
  4:        100,         720,         873,         288,         25;
  5:       1700,       12640,       16461,        6448,        875,        36;
  6:      44200,      335440,      456426,      196912,      32825,      2196,       49;
  7:    1635400,    12588080,    17642502,     8097168,    1522200,    128520,     4802,      64;
  8:   81770000,   635945600,   910448280,   436222848,   88761825,   8617968,   415030,    9472,    81;
  9: 5315050000, 41663544000, 60610015800, 29973059840, 6451116825, 687984948, 38706962, 1157760, 17253, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-8*n^3+26*n^2-40*n+25+(4*n^2-16*n+20)*x-9*x^2)*P(n-2,x) + ((4*n^2-16*n+20)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+7+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x - 9*x^2)*P[n - 2, x] + ((4*n^2 - 16*n + 20)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 7 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x - 9*x^2)*v[i - 2] + ((4*n^2 - 16*n + 20)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 7 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], x)); 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. [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.