Real-rooted atlas

A143495 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,n + n^{2}\right)\,P_{n}(x) = \left(-2 - 3\,n - 12\,n x + 16\,n x^{2} - n^{2} - 12\,n^{2} x + 41\,n^{2} x^{2} - 4\,n^{3} x + 34\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-x + 13\,x^{2} - 23\,x^{3} + 20\,n x^{2} - 52\,n x^{3} + 16\,n^{2} x^{2} - 31\,n^{2} x^{3} + 4\,n^{3} x^{2} + 2\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(x^{3} - 7\,x^{4} - 18\,n x^{4} - 15\,n^{2} x^{4} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(3 + 4\,x + 5\,n + 12\,n x + 2\,n^{2} + 12\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(x - 8\,x^{2} - 20\,n x^{2} - 16\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211213
312881110
4148333432814
5172873280020255771
6110018451056019525874840779
7113234203024010202512121235721292751
811685796728003855258557926956531399682155703

All 15 rows

Technical details
Node id
n-dba3a10a7cd587d6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-119b729771ac2c5d

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  12;
   3: 1,  28,    81;
   4: 1,  48,   333,    432;
   5: 1,  72,   873,   2800,    2025;
   6: 1, 100,  1845,  10560,   19525,     8748;
   7: 1, 132,  3420,  30240,  102025,   121212,    35721;
   8: 1, 168,  5796,  72800,  385525,   855792,   695653,   139968;
   9: 1, 208,  9198, 155232, 1181775,  4186728,  6519793,  3774400,   531441;
  10: 1, 252, 13878, 302400, 3122175, 16099020, 40711405, 46352640, 19639341, 1968300;
%F Formula
The row polynomials satisfy (n^2+2*n+1)*P(n,x) = -(n^2+3*n+2+(4*n^3+12*n^2+12*n)*x-(9*n^4+34*n^3+41*n^2+16*n)*x^2)*P(n-2,x) - (x-(4*n^3+16*n^2+20*n+13)*x^2-(4*n^4+2*n^3-31*n^2-52*n-23)*x^3)*d/dx P(n-2,x) + (x^3-(4*n^3+15*n^2+18*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+5*n+3+(4*n^3+12*n^2+12*n+4)*x)*P(n-1,x) + (x-(4*n^3+16*n^2+20*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (4*n^3 + 12*n^2 + 12*n)*x - (9*n^4 + 34*n^3 + 41*n^2 + 16*n)*x^2)*P[n - 2, x] - (x - (4*n^3 + 16*n^2 + 20*n + 13)*x^2 - (4*n^4 + 2*n^3 - 31*n^2 - 52*n - 23)*x^3)*D[P[n - 2, x], x] + (x^3 - (4*n^3 + 15*n^2 + 18*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 5*n + 3 + (4*n^3 + 12*n^2 + 12*n + 4)*x)*P[n - 1, x] + (x - (4*n^3 + 16*n^2 + 20*n + 8)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 12*x; for(n = 3, N, my(i = n); v[i] = (-(n^2 + 3*n + 2 + (4*n^3 + 12*n^2 + 12*n)*x - (9*n^4 + 34*n^3 + 41*n^2 + 16*n)*x^2)*v[i - 2] - (x - (4*n^3 + 16*n^2 + 20*n + 13)*x^2 - (4*n^4 + 2*n^3 - 31*n^2 - 52*n - 23)*x^3)*deriv(v[i - 2], x) + (x^3 - (4*n^3 + 15*n^2 + 18*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 5*n + 3 + (4*n^3 + 12*n^2 + 12*n + 4)*x)*v[i - 1] + (x - (4*n^3 + 16*n^2 + 20*n + 8)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.