Real-rooted atlas

A131689 → (θ+1)² → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(7\,x + 29\,x^{2} + 45\,x^{3} + 31\,x^{4} + 8\,x^{5} - n x - 5\,n x^{2} - 9\,n x^{3} - 7\,n x^{4} - 2\,n x^{5}\right)\,P_{n-2}(x) + \left(x + 6\,x^{2} + 14\,x^{3} + 16\,x^{4} + 9\,x^{5} + 2\,x^{6}\right)\,P_{n-2}'(x) + \left(-1 + 2\,x + 16\,x^{2} + 26\,x^{3} + 17\,x^{4} + 4\,x^{5} - n x - 5\,n x^{2} - 9\,n x^{3} - 7\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}(x) + \left(x + 6\,x^{2} + 14\,x^{3} + 16\,x^{4} + 9\,x^{5} + 2\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2044
3042226
40462154220
50413884013062288
6042863234116261299428144
7045781091268308170972148282399056
8041158346263370041419576270175819082746402400

All 15 rows

Technical details
Node id
n-4a382eb0a4874d05 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3731e98b1393eb57

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: 0, 4;
   3: 0, 4,   22;
   4: 0, 4,   62,    154;
   5: 0, 4,  138,    840,    1306;
   6: 0, 4,  286,   3234,   11626,    12994;
   7: 0, 4,  578,  10912,   68308,   170972,    148282;
   8: 0, 4, 1158,  34626,  337004,  1419576,   2701758,   1908274;
   9: 0, 4, 2314, 106536, 1518710,  9591620,  30077814,  45968464,  27333706;
  10: 0, 4, 4622, 322642, 6501014, 57855250, 266053034, 660317782, 840869042, 431220034;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -((n-7)*x+(5*n-29)*x^2+(9*n-45)*x^3+(7*n-31)*x^4+(2*n-8)*x^5)*P(n-2,x) + (x+6*x^2+14*x^3+16*x^4+9*x^5+2*x^6)*d/dx P(n-2,x) - (1+(n-2)*x+(5*n-16)*x^2+(9*n-26)*x^3+(7*n-17)*x^4+(2*n-4)*x^5)*P(n-1,x) + (x+6*x^2+14*x^3+16*x^4+9*x^5+2*x^6)*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[Together[(-((n - 7)*x + (5*n - 29)*x^2 + (9*n - 45)*x^3 + (7*n - 31)*x^4 + (2*n - 8)*x^5)*P[n - 2, x] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*D[P[n - 2, x], x] - (1 + (n - 2)*x + (5*n - 16)*x^2 + (9*n - 26)*x^3 + (7*n - 17)*x^4 + (2*n - 4)*x^5)*P[n - 1, x] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
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] = (-((n - 7)*x + (5*n - 29)*x^2 + (9*n - 45)*x^3 + (7*n - 31)*x^4 + (2*n - 8)*x^5)*v[i - 2] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*deriv(v[i - 2], x) - (1 + (n - 2)*x + (5*n - 16)*x^2 + (9*n - 26)*x^3 + (7*n - 17)*x^4 + (2*n - 4)*x^5)*v[i - 1] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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. [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.