Real-rooted atlas

A169803 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - n + 8\,n x + n^{2} - 3\,n^{2} x - n^{2} x^{2} + n^{3} - 6\,n^{3} x + 3\,n^{3} x^{2} + 3\,n^{4} x - 3\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-16\,x + 16\,x^{2} + n - 22\,n x + 12\,n x^{2} + 2\,n^{2} - 6\,n^{2} x - 26\,n^{2} x^{2} - 4\,n^{2} x^{3} + n^{3} + 5\,n^{3} x - 11\,n^{3} x^{2} - 6\,n^{3} x^{3} + 3\,n^{4} x + 8\,n^{4} x^{2} + n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-x - 2\,x^{2} + 8\,x^{3} - 2\,n x - 6\,n x^{2} + 8\,n x^{3} - n^{2} x - 7\,n^{2} x^{2} - 13\,n^{2} x^{3} - 4\,n^{2} x^{4} - 3\,n^{3} x^{2} - 14\,n^{3} x^{3} - 8\,n^{3} x^{4} - n^{4} x^{3} - 4\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
222
33811
442428
55482780
6680108194
7712027064461
881685403201036
992249459601252263
1010288151222407504800
11113602268448026252169960
1212440324080647000151220268
131352844551344015750604834340577
14146245940211203150018144274480086
Technical details
Node id
n-afcfdc89e46e8d20 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-66e6f7e007ec6615

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 0;
  1: 1;
  2: 2;
  3: 3,   8;
  4: 4,  24;
  5: 5,  48,  27;
  6: 6,  80, 108;
  7: 7, 120, 270,  64;
  8: 8, 168, 540, 320;
  9: 9, 224, 945, 960, 125;
%F Formula
The row polynomials satisfy (n^3+n^2-n-1+(3*n^4-6*n^3-3*n^2+8*n-2)*x+(n^5-3*n^4+3*n^3-n^2)*x^2)*P(n,x) = (n^3+2*n^2+n+(3*n^4+5*n^3-6*n^2-22*n-16)*x+(n^5+8*n^4-11*n^3-26*n^2+12*n+16)*x^2+(2*n^5-6*n^3-4*n^2)*x^3)*P(n-1,x) - ((n^2+2*n+1)*x+(3*n^3+7*n^2+6*n+2)*x^2+(n^4+14*n^3+13*n^2-8*n-8)*x^3+(4*n^4+8*n^3+4*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 2*n^2 + n + (3*n^4 + 5*n^3 - 6*n^2 - 22*n - 16)*x + (n^5 + 8*n^4 - 11*n^3 - 26*n^2 + 12*n + 16)*x^2 + (2*n^5 - 6*n^3 - 4*n^2)*x^3)*P[n - 1, x] - ((n^2 + 2*n + 1)*x + (3*n^3 + 7*n^2 + 6*n + 2)*x^2 + (n^4 + 14*n^3 + 13*n^2 - 8*n - 8)*x^3 + (4*n^4 + 8*n^3 + 4*n^2)*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 - n - 1 + (3*n^4 - 6*n^3 - 3*n^2 + 8*n - 2)*x + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 2*n^2 + n + (3*n^4 + 5*n^3 - 6*n^2 - 22*n - 16)*x + (n^5 + 8*n^4 - 11*n^3 - 26*n^2 + 12*n + 16)*x^2 + (2*n^5 - 6*n^3 - 4*n^2)*x^3)*v[i - 1] - ((n^2 + 2*n + 1)*x + (3*n^3 + 7*n^2 + 6*n + 2)*x^2 + (n^4 + 14*n^3 + 13*n^2 - 8*n - 8)*x^3 + (4*n^4 + 8*n^3 + 4*n^2)*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 - n - 1 + (3*n^4 - 6*n^3 - 3*n^2 + 8*n - 2)*x + (n^5 - 3*n^4 + 3*n^3 - n^2)*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

Same fingerprint ?

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