Real-rooted atlas

A008277 → (θ+1)² → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 + 32\,x - 6\,x^{2} + 4\,n - 24\,n x + 5\,n x^{2} + 4\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 28\,x^{2} - 6\,x^{3} - 8\,n x^{2} + 2\,n x^{3}\right)\,P_{n-2}'(x) + \left(4\,x^{3} - x^{4}\right)\,P_{n-2}''(x) + \left(12 - 3\,x - 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(4\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2415
3912122
4165428199
525160225601471
636375104081012412386
749756350056002709252112867
8641372957626250272168694508173681

All 15 rows

Technical details
Node id
n-435207fe015f8c82 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-915ca4b22aed877c

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:   4,    1;
   3:   9,   12,     1;
   4:  16,   54,    28,      1;
   5:  25,  160,   225,     60,      1;
   6:  36,  375,  1040,    810,    124,       1;
   7:  49,  756,  3500,   5600,   2709,     252,      1;
   8:  64, 1372,  9576,  26250,  27216,    8694,    508,     1;
   9:  81, 2304, 22638,  95256, 173775,  124320,  27225,  1020,    1;
  10: 100, 3645, 48000, 288120, 821772, 1063125, 545680, 83970, 2044, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-3+(4*n^2-24*n+32)*x-(n^2-5*n+6)*x^2)*P(n-2,x) - (4*x+(8*n-28)*x^2-(2*n-6)*x^3)*d/dx P(n-2,x) + (4*x^3-x^4)*d^2/dx^2 P(n-2,x) - (4*n-12-(2*n-3)*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[(4*n - 3 + (4*n^2 - 24*n + 32)*x - (n^2 - 5*n + 6)*x^2)*P[n - 2, x] - (4*x + (8*n - 28)*x^2 - (2*n - 6)*x^3)*D[P[n - 2, x], x] + (4*x^3 - x^4)*D[P[n - 2, x], {x, 2}] - (4*n - 12 - (2*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + x; for(n = 3, N, my(i = n); v[i] = (4*n - 3 + (4*n^2 - 24*n + 32)*x - (n^2 - 5*n + 6)*x^2)*v[i - 2] - (4*x + (8*n - 28)*x^2 - (2*n - 6)*x^3)*deriv(v[i - 2], x) + (4*x^3 - x^4)*deriv(deriv(v[i - 2], x), x) - (4*n - 12 - (2*n - 3)*x)*v[i - 1] + (4*x - 2*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.