Real-rooted atlas

A080248 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\right)\,P_{n}(x) = \left(-8\,x + 36\,x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 26\,x^{2} + 16\,x^{3}\right)\,P_{n-2}'(x) + \left(-6\,x^{2} + 42\,x^{3}\right)\,P_{n-2}''(x) + \left(-x^{3} + 8\,x^{4}\right)\,P_{n-2}'''(x) + \left(4 + 16\,x\right)\,P_{n-1}(x) + \left(10\,x - 16\,x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
2116926
31529016159
41160657320251163
51484430243688753610066
611456269015132819950201649101701
714372164682561104379450710644116641184853

All 15 rows

Technical details
Node id
n-9f0689b7b1ce4764 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-87f7b736a7649910

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: 1,    16,       9;
  3: 1,    52,      90,       16;
  4: 1,   160,     657,      320,        25;
  5: 1,   484,    4302,     4368,       875,       36;
  6: 1,  1456,   26901,    51328,     19950,     2016,      49;
  7: 1,  4372,  164682,   561104,    379450,    71064,    4116,     64;
  8: 1, 13120,  997929,  5903808,   6568475,  2038752,  211974,   7680,    81;
  9: 1, 39364, 6017094, 60812176, 107751825, 52272396, 8710240, 553344, 13365, 100;
%F Formula
The row polynomials satisfy 4*P(n,x) = -(8*x-36*x^2)*P(n-2,x) - (6*x-26*x^2-16*x^3)*d/dx P(n-2,x) - (6*x^2-42*x^3)*d^2/dx^2 P(n-2,x) - (x^3-8*x^4)*d^3/dx^3 P(n-2,x) + (4+16*x)*P(n-1,x) + (10*x-16*x^2)*d/dx P(n-1,x) + 2*x^2*d^2/dx^2 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[Together[(-(8*x - 36*x^2)*P[n - 2, x] - (6*x - 26*x^2 - 16*x^3)*D[P[n - 2, x], x] - (6*x^2 - 42*x^3)*D[P[n - 2, x], {x, 2}] - (x^3 - 8*x^4)*D[P[n - 2, x], {x, 3}] + (4 + 16*x)*P[n - 1, x] + (10*x - 16*x^2)*D[P[n - 1, x], x] + 2*x^2*D[P[n - 1, x], {x, 2}])/(4)]];
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] = (-(8*x - 36*x^2)*v[i - 2] - (6*x - 26*x^2 - 16*x^3)*deriv(v[i - 2], x) - (6*x^2 - 42*x^3)*deriv(deriv(v[i - 2], x), x) - (x^3 - 8*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (4 + 16*x)*v[i - 1] + (10*x - 16*x^2)*deriv(v[i - 1], x) + 2*x^2*deriv(deriv(v[i - 1], x), x))/(4)); 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.