Real-rooted atlas

A121574 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-n + 4\,n x + 8\,n x^{2} - 3\,n^{2} x + 6\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8\,x + 28\,x^{2} + 32\,x^{3} - n - 8\,n x + 2\,n x^{2} + 24\,n x^{3} - 3\,n^{2} x - 12\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(x + 12\,x^{2} + 36\,x^{3} + 16\,x^{4} + 3\,n x^{2} + 24\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
21101223
3116483297
412211117680390
51282015365601921518
61343181220212016324485773
71404622336584073924480102421575

All 15 rows

Technical details
Node id
n-609743fe373f06f1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-552b265c5cde19a5

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, 10,  12;
  3: 1, 16,  48,   32;
  4: 1, 22, 111,  176,    80;
  5: 1, 28, 201,  536,   560,   192;
  6: 1, 34, 318, 1220,  2120,  1632,   448;
  7: 1, 40, 462, 2336,  5840,  7392,  4480,  1024;
  8: 1, 46, 633, 3992, 13205, 23952, 23632, 11776,  2304;
  9: 1, 52, 831, 6296, 26105, 62652, 88144, 70912, 29952, 5120;
%F Formula
The row polynomials satisfy -(n+(3*n^2-4*n)*x-(6*n^2+8*n)*x^2)*P(n,x) = -(n+(3*n^2+8*n-8)*x+(12*n^2-2*n-28)*x^2-(24*n+32)*x^3)*P(n-1,x) + (x+(3*n+12)*x^2+(24*n+36)*x^3+(12*n+16)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (3*n^2 + 8*n - 8)*x + (12*n^2 - 2*n - 28)*x^2 - (24*n + 32)*x^3)*P[n - 1, x] + (x + (3*n + 12)*x^2 + (24*n + 36)*x^3 + (12*n + 16)*x^4)*D[P[n - 1, x], x])/(-n - (3*n^2 - 4*n)*x + (6*n^2 + 8*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n + (3*n^2 + 8*n - 8)*x + (12*n^2 - 2*n - 28)*x^2 - (24*n + 32)*x^3)*v[i - 1] + (x + (3*n + 12)*x^2 + (24*n + 36)*x^3 + (12*n + 16)*x^4)*deriv(v[i - 1], x))/(-n - (3*n^2 - 4*n)*x + (6*n^2 + 8*n)*x^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.