Real-rooted atlas

A052553 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-3 + 66\,x - 108\,x^{2} + 7\,n - 148\,n x + 216\,n x^{2} - 5\,n^{2} + 105\,n^{2} x - 171\,n^{2} x^{2} + n^{3} - 30\,n^{3} x + 67\,n^{3} x^{2} + 3\,n^{4} x - 13\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-2 + 12\,x + 72\,x^{2} - 32\,x^{3} + 5\,n - 34\,n x - 192\,n x^{2} + 104\,n x^{3} - 4\,n^{2} + 36\,n^{2} x + 152\,n^{2} x^{2} - 128\,n^{2} x^{3} + n^{3} - 19\,n^{3} x - 35\,n^{3} x^{2} + 74\,n^{3} x^{3} + 3\,n^{4} x - 2\,n^{4} x^{2} - 20\,n^{4} x^{3} + n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-x + 6\,x^{2} + 36\,x^{3} - 16\,x^{4} + 2\,n x - 14\,n x^{2} - 76\,n x^{3} + 48\,n x^{4} - n^{2} x + 11\,n^{2} x^{2} + 47\,n^{2} x^{3} - 52\,n^{2} x^{4} - 3\,n^{3} x^{2} - 6\,n^{3} x^{3} + 24\,n^{3} x^{4} - n^{4} x^{3} - 4\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 1\)

Rows ?

nx0x1x2Pn(1)
10
20
311
422
53811
642428
75482780
8680108194

All 15 rows

Technical details
Node id
n-27abf076aa58a72b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-66e6f7e007ec6615

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -3)
  • 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: 0;
   2: 0;
   3: 1;
   4: 2;
   5: 3,   8;
   6: 4,  24;
   7: 5,  48,  27;
   8: 6,  80, 108;
   9: 7, 120, 270,  64;
  10: 8, 168, 540, 320;
%F Formula
The row polynomials satisfy (n^3-5*n^2+7*n-3+(3*n^4-30*n^3+105*n^2-148*n+66)*x+(n^5-13*n^4+67*n^3-171*n^2+216*n-108)*x^2)*P(n,x) = (n^3-4*n^2+5*n-2+(3*n^4-19*n^3+36*n^2-34*n+12)*x+(n^5-2*n^4-35*n^3+152*n^2-192*n+72)*x^2+(2*n^5-20*n^4+74*n^3-128*n^2+104*n-32)*x^3)*P(n-1,x) - ((n^2-2*n+1)*x+(3*n^3-11*n^2+14*n-6)*x^2+(n^4+6*n^3-47*n^2+76*n-36)*x^3+(4*n^4-24*n^3+52*n^2-48*n+16)*x^4)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 4*n^2 + 5*n - 2 + (3*n^4 - 19*n^3 + 36*n^2 - 34*n + 12)*x + (n^5 - 2*n^4 - 35*n^3 + 152*n^2 - 192*n + 72)*x^2 + (2*n^5 - 20*n^4 + 74*n^3 - 128*n^2 + 104*n - 32)*x^3)*P[n - 1, x] - ((n^2 - 2*n + 1)*x + (3*n^3 - 11*n^2 + 14*n - 6)*x^2 + (n^4 + 6*n^3 - 47*n^2 + 76*n - 36)*x^3 + (4*n^4 - 24*n^3 + 52*n^2 - 48*n + 16)*x^4)*D[P[n - 1, x], x])/(n^3 - 5*n^2 + 7*n - 3 + (3*n^4 - 30*n^3 + 105*n^2 - 148*n + 66)*x + (n^5 - 13*n^4 + 67*n^3 - 171*n^2 + 216*n - 108)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = ((n^3 - 4*n^2 + 5*n - 2 + (3*n^4 - 19*n^3 + 36*n^2 - 34*n + 12)*x + (n^5 - 2*n^4 - 35*n^3 + 152*n^2 - 192*n + 72)*x^2 + (2*n^5 - 20*n^4 + 74*n^3 - 128*n^2 + 104*n - 32)*x^3)*v[i - 1] - ((n^2 - 2*n + 1)*x + (3*n^3 - 11*n^2 + 14*n - 6)*x^2 + (n^4 + 6*n^3 - 47*n^2 + 76*n - 36)*x^3 + (4*n^4 - 24*n^3 + 52*n^2 - 48*n + 16)*x^4)*deriv(v[i - 1], x))/(n^3 - 5*n^2 + 7*n - 3 + (3*n^4 - 30*n^3 + 105*n^2 - 148*n + 66)*x + (n^5 - 13*n^4 + 67*n^3 - 171*n^2 + 216*n - 108)*x^2)); 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. [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.