Real-rooted atlas

A008292 → θ+1 (shift +2) → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(-2\,x^{2} - 6\,x^{3} - 6\,x^{4} - 2\,x^{5} - n x^{2} - 3\,n x^{3} - 3\,n x^{4} - n x^{5}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 8\,x^{3} - 12\,x^{4} - 8\,x^{5} - 2\,x^{6} - n x^{2} - 4\,n x^{3} - 6\,n x^{4} - 4\,n x^{5} - n x^{6}\right)\,P_{n-2}'(x) + \left(1 + 6\,x + 12\,x^{2} + 10\,x^{3} + 3\,x^{4} + n x + 3\,n x^{2} + 3\,n x^{3} + n x^{4}\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} + 6\,x^{3} + 4\,x^{4} + x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 10\,x + 12\,x^{2}\)
\(P_{2}(x) = 1 + 25\,x + 80\,x^{2} + 60\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11101223
21258060166
31563606603601437
41119137246205880252014512
51246478826376588005712020160167491
61501158641343164667047711206048001814402174746
7110125088063702032382008094240105840006955200181440031374953
812035159764288222020603880730976401417046401530144008648640019958400497909380

All 15 rows

Technical details
Node id
n-82a92db852913f5a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-900195978027e341

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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,   10,      12;
   2: 1,   25,      80,       60;
   3: 1,   56,     360,      660,       360;
   4: 1,  119,    1372,     4620,      5880,       2520;
   5: 1,  246,    4788,    26376,     58800,      57120,       20160;
   6: 1,  501,   15864,   134316,    466704,     771120,      604800,      181440;
   7: 1, 1012,   50880,   637020,   3238200,    8094240,    10584000,     6955200,     1814400;
   8: 1, 2035,  159764,  2882220,  20603880,   73097640,   141704640,   153014400,    86486400,    19958400;
   9: 1, 4082,  494604, 12623952, 123552000,  597738240,  1611308160,  2542034880,  2335132800,  1157587200,   239500800;
  10: 1, 8177, 1516528, 54031692, 710231808, 4559480640, 16416354240, 35485369920, 47100493440, 37621584000, 16605388800, 3113510400;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -((n+2)*x^2+(3*n+6)*x^3+(3*n+6)*x^4+(n+2)*x^5)*P(n-2,x) - ((n+2)*x^2+(4*n+8)*x^3+(6*n+12)*x^4+(4*n+8)*x^5+(n+2)*x^6)*d/dx P(n-2,x) + (1+(n+6)*x+(3*n+12)*x^2+(3*n+10)*x^3+(n+3)*x^4)*P(n-1,x) + (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 10*x + 12*x^2, P(2,x) = 1 + 25*x + 80*x^2 + 60*x^3.
%t Mathematica
P[1, x_] := 1 + 10*x + 12*x^2;
P[2, x_] := 1 + 25*x + 80*x^2 + 60*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((n + 2)*x^2 + (3*n + 6)*x^3 + (3*n + 6)*x^4 + (n + 2)*x^5)*P[n - 2, x] - ((n + 2)*x^2 + (4*n + 8)*x^3 + (6*n + 12)*x^4 + (4*n + 8)*x^5 + (n + 2)*x^6)*D[P[n - 2, x], x] + (1 + (n + 6)*x + (3*n + 12)*x^2 + (3*n + 10)*x^3 + (n + 3)*x^4)*P[n - 1, x] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*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 + 10*x + 12*x^2; v[2] = 1 + 25*x + 80*x^2 + 60*x^3; for(n = 3, N, my(i = n); v[i] = (-((n + 2)*x^2 + (3*n + 6)*x^3 + (3*n + 6)*x^4 + (n + 2)*x^5)*v[i - 2] - ((n + 2)*x^2 + (4*n + 8)*x^3 + (6*n + 12)*x^4 + (4*n + 8)*x^5 + (n + 2)*x^6)*deriv(v[i - 2], x) + (1 + (n + 6)*x + (3*n + 12)*x^2 + (3*n + 10)*x^3 + (n + 3)*x^4)*v[i - 1] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.