Real-rooted atlas

A105070 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x - n^{2} x - 5\,n^{3} x + n^{3} x^{2} + 2\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(1 - 6\,x + 12\,x^{2} - 8\,x^{3} + n + 9\,n x + 26\,n x^{2} - 16\,n x^{3} - 6\,n^{2} x - 19\,n^{2} x^{2} - 2\,n^{2} x^{3} - 5\,n^{3} x - 15\,n^{3} x^{2} + 14\,n^{3} x^{3} + 3\,n^{4} x^{2} + 10\,n^{4} x^{3} + n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - 12\,x^{2} + 24\,x^{3} - 16\,x^{4} - 22\,n x^{2} + 68\,n x^{3} - 48\,n x^{4} - 10\,n^{2} x^{2} + 46\,n^{2} x^{3} - 52\,n^{2} x^{4} + 12\,n^{3} x^{3} - 24\,n^{3} x^{4} + 2\,n^{4} x^{3} - 4\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
23811
343236
458036121
56160216382
672807561281171
78448201610243496
896724536460840010225
910960907215360400029402
10111320166324224022000115283355
11121760285121013768800013824233484
1213228846332219648286000898563136647273
1314291272072439296800800419328439041778326
141536401081088236802002000157248032928081924847395
Technical details
Node id
n-49db29a6af8e769f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2a24e167d1d25e8a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:  2;
  2:  3,   8;
  3:  4,  32;
  4:  5,  80,   36;
  5:  6, 160,  216;
  6:  7, 280,  756,   128;
  7:  8, 448, 2016,  1024;
  8:  9, 672, 4536,  4608,  400;
  9: 10, 960, 9072, 15360, 4000;
%F Formula
The row polynomials satisfy (n-(5*n^3+n^2-2*n)*x+(n^5+2*n^4+n^3)*x^2)*P(n,x) = (n+1-(5*n^3+6*n^2-9*n+6)*x+(n^5+3*n^4-15*n^3-19*n^2+26*n+12)*x^2+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^3)*P(n-1,x) + (2*x-(10*n^2+22*n+12)*x^2+(2*n^4+12*n^3+46*n^2+68*n+24)*x^3-(4*n^4+24*n^3+52*n^2+48*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 + 1 - (5*n^3 + 6*n^2 - 9*n + 6)*x + (n^5 + 3*n^4 - 15*n^3 - 19*n^2 + 26*n + 12)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*P[n - 1, x] + (2*x - (10*n^2 + 22*n + 12)*x^2 + (2*n^4 + 12*n^3 + 46*n^2 + 68*n + 24)*x^3 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*D[P[n - 1, x], x])/(n - (5*n^3 + n^2 - 2*n)*x + (n^5 + 2*n^4 + n^3)*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 + 1 - (5*n^3 + 6*n^2 - 9*n + 6)*x + (n^5 + 3*n^4 - 15*n^3 - 19*n^2 + 26*n + 12)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*v[i - 1] + (2*x - (10*n^2 + 22*n + 12)*x^2 + (2*n^4 + 12*n^3 + 46*n^2 + 68*n + 24)*x^3 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*deriv(v[i - 1], x))/(n - (5*n^3 + n^2 - 2*n)*x + (n^5 + 2*n^4 + n^3)*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.