Real-rooted atlas

A008277 → θ+1 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-1 - x + 6\,x^{2} + 14\,x^{3} + 11\,x^{4} + 3\,x^{5} - n x - 4\,n x^{2} - 6\,n x^{3} - 4\,n x^{4} - n x^{5}\right)\,P_{n-2}(x) + \left(x^{2} + 4\,x^{3} + 6\,x^{4} + 4\,x^{5} + x^{6}\right)\,P_{n-2}'(x) + \left(2 + 5\,x + 2\,x^{2} - 3\,x^{3} - 2\,x^{4} + n x + 3\,n x^{2} + 3\,n x^{3} + n x^{4}\right)\,P_{n-1}(x) + \left(-x^{2} - 3\,x^{3} - 3\,x^{4} - x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
33121025
44306237133
5560225320151761
66105620156017126744677
7716814355600106689604326330745
8825229401645047964738785667417007215173

All 15 rows

Technical details
Node id
n-53e8f91758e13abb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5d1081b0b48f32c6

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:  2,   3;
   3:  3,  12,   10;
   4:  4,  30,   62,    37;
   5:  5,  60,  225,   320,    151;
   6:  6, 105,  620,  1560,   1712,     674;
   7:  7, 168, 1435,  5600,  10668,    9604,    3263;
   8:  8, 252, 2940, 16450,  47964,   73878,   56674,   17007;
   9:  9, 360, 5502, 41832, 173355,  404124,  524415,  351792,   94828;
  10: 10, 495, 9600, 95340, 533862, 1754445, 3413260, 3837720, 2293862, 562595;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(1+(n+1)*x+(4*n-6)*x^2+(6*n-14)*x^3+(4*n-11)*x^4+(n-3)*x^5)*P(n-2,x) + (x^2+4*x^3+6*x^4+4*x^5+x^6)*d/dx P(n-2,x) + (2+(n+5)*x+(3*n+2)*x^2+(3*n-3)*x^3+(n-2)*x^4)*P(n-1,x) - (x^2+3*x^3+3*x^4+x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + (n + 1)*x + (4*n - 6)*x^2 + (6*n - 14)*x^3 + (4*n - 11)*x^4 + (n - 3)*x^5)*P[n - 2, x] + (x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*D[P[n - 2, x], x] + (2 + (n + 5)*x + (3*n + 2)*x^2 + (3*n - 3)*x^3 + (n - 2)*x^4)*P[n - 1, x] - (x^2 + 3*x^3 + 3*x^4 + x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 3*x; for(n = 3, N, my(i = n); v[i] = (-(1 + (n + 1)*x + (4*n - 6)*x^2 + (6*n - 14)*x^3 + (4*n - 11)*x^4 + (n - 3)*x^5)*v[i - 2] + (x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*deriv(v[i - 2], x) + (2 + (n + 5)*x + (3*n + 2)*x^2 + (3*n - 3)*x^3 + (n - 2)*x^4)*v[i - 1] - (x^2 + 3*x^3 + 3*x^4 + x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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. [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.