Real-rooted atlas

A112486 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
22202749
36104315240665
42461630605040262511365
5120417629736784008662534020233077
6720321123036601128512196875016216205093555564729
750402770563310956161436803928925050644440331080758648640151427137

All 15 rows

Technical details
Node id
n-e723da369b611fa9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-48e7401d7923c889

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,       20,        27;
  3:      6,      104,       315,        240;
  4:     24,      616,      3060,       5040,        2625;
  5:    120,     4176,     29736,      78400,       86625,       34020;
  6:    720,    32112,    303660,    1128512,     1968750,     1621620,      509355;
  7:   5040,   277056,   3310956,   16143680,    39289250,    50644440,    33108075,     8648640;
  8:  40320,  2654784,  38719944,  236441920,   748940500,  1337295960,  1359638280,   735134400,   164189025;
  9: 362880, 27999360, 485652024, 3594493056, 14169512500, 32742309600, 45877491660, 38399961600, 17677685025, 3445942500;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(4*n^2-4*n)*x-(9*n^2-16*n+7)*x^2)*P(n-2,x) - ((2*n-1)*x-(4*n^2-8*n)*x^2-(4*n^2+6*n-7)*x^3)*d/dx P(n-2,x) - (x^2-(4*n-4)*x^3-(4*n-3)*x^4)*d^2/dx^2 P(n-2,x) + (2*n-1+4*n*x)*P(n-1,x) + (2*x-(4*n-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^2 - 2*n + 1 + (4*n^2 - 4*n)*x - (9*n^2 - 16*n + 7)*x^2)*P[n - 2, x] - ((2*n - 1)*x - (4*n^2 - 8*n)*x^2 - (4*n^2 + 6*n - 7)*x^3)*D[P[n - 2, x], x] - (x^2 - (4*n - 4)*x^3 - (4*n - 3)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n - 1 + 4*n*x)*P[n - 1, x] + (2*x - (4*n - 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^2 - 2*n + 1 + (4*n^2 - 4*n)*x - (9*n^2 - 16*n + 7)*x^2)*v[i - 2] - ((2*n - 1)*x - (4*n^2 - 8*n)*x^2 - (4*n^2 + 6*n - 7)*x^3)*deriv(v[i - 2], x) - (x^2 - (4*n - 4)*x^3 - (4*n - 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n - 1 + 4*n*x)*v[i - 1] + (2*x - (4*n - 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.