Real-rooted atlas

A098158 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-12\,x + 2\,n + 26\,n x + n^{2} - 18\,n^{2} x - n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(4\,x + 8\,x^{2} - 2\,n + 4\,n x + 24\,n x^{2} - n^{2} + 11\,n^{2} x - 38\,n^{2} x^{2} + n^{3} - 7\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2} + 4\,x^{3} - 2\,n x - 6\,n x^{2} + 8\,n x^{3} - 2\,n^{2} x + 10\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2022
30235
4009413
500324532
60002050676
7000475907176
80000352101478400
9000051684902249896
1000000545881008324101984
1100000631516801890450114352
1200000077132041583300605129472
1300000075284455924054457921320480
14000000010425741287018876858010141444032
15000000088191001033033360361301312741594208
Technical details
Node id
n-7ef9e1442c0e1a16 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6116eab11eb14fe9

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, 2;
   3: 0, 2, 3;
   4: 0, 0, 9,  4;
   5: 0, 0, 3, 24,  5;
   6: 0, 0, 0, 20, 50,   6;
   7: 0, 0, 0,  4, 75,  90,   7;
   8: 0, 0, 0,  0, 35, 210, 147,    8;
   9: 0, 0, 0,  0,  5, 168, 490,  224,   9;
  10: 0, 0, 0,  0,  0,  54, 588, 1008, 324, 10;
%F Formula
The row polynomials satisfy -(n^3-n^2-2*n-(4*n^3-18*n^2+26*n-12)*x)*P(n,x) = (n^3-n^2-2*n-(7*n^3-11*n^2-4*n-4)*x+(12*n^3-38*n^2+24*n+8)*x^2)*P(n-1,x) - ((2*n^2+2*n)*x-(10*n^2-6*n-4)*x^2+(8*n^2-8*n-4)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 - 2*n - (7*n^3 - 11*n^2 - 4*n - 4)*x + (12*n^3 - 38*n^2 + 24*n + 8)*x^2)*P[n - 1, x] - ((2*n^2 + 2*n)*x - (10*n^2 - 6*n - 4)*x^2 + (8*n^2 - 8*n - 4)*x^3)*D[P[n - 1, x], x])/(-n^3 + n^2 + 2*n + (4*n^3 - 18*n^2 + 26*n - 12)*x)]];
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*x; for(n = 3, N, my(i = n); v[i] = ((n^3 - n^2 - 2*n - (7*n^3 - 11*n^2 - 4*n - 4)*x + (12*n^3 - 38*n^2 + 24*n + 8)*x^2)*v[i - 1] - ((2*n^2 + 2*n)*x - (10*n^2 - 6*n - 4)*x^2 + (8*n^2 - 8*n - 4)*x^3)*deriv(v[i - 1], x))/(-n^3 + n^2 + 2*n + (4*n^3 - 18*n^2 + 26*n - 12)*x)); 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.