Real-rooted atlas

A199478 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 8\,x + 48\,x^{2} - 3\,n - 4\,n x + 52\,n x^{2} - n^{2} - 8\,n^{2} x + 18\,n^{2} x^{2} - 2\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-2 + 4\,x + 64\,x^{2} + 96\,x^{3} - 3\,n - 22\,n x + 26\,n x^{2} + 152\,n x^{3} - n^{2} - 16\,n^{2} x - 20\,n^{2} x^{2} + 64\,n^{2} x^{3} - 2\,n^{3} x - 6\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 26\,x^{2} + 56\,x^{3} - 192\,x^{4} + n x + 18\,n x^{2} + 66\,n x^{3} - 112\,n x^{4} + 2\,n^{2} x^{2} + 14\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4Pn(1)
0123
1167
2110617
31142439
4118541689
51229680199
612615022440441
7130216480240967

All 15 rows

Technical details
Node id
n-4aeccb7fad8daae1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6bd563b74ee6c5a6

Observed on the stored rows ?

  • Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1,  2;
  1: 1,  6;
  2: 1, 10,   6;
  3: 1, 14,  24;
  4: 1, 18,  54,   16;
  5: 1, 22,  96,   80;
  6: 1, 26, 150,  224,   40;
  7: 1, 30, 216,  480,  240;
  8: 1, 34, 294,  880,  800,  96;
  9: 1, 38, 384, 1456, 2000, 672;
%F Formula
The row polynomials satisfy -(n^2+3*n+2+(2*n^3+8*n^2+4*n-8)*x-(2*n^3+18*n^2+52*n+48)*x^2)*P(n,x) = -(n^2+3*n+2+(2*n^3+16*n^2+22*n-4)*x+(6*n^3+20*n^2-26*n-64)*x^2-(8*n^3+64*n^2+152*n+96)*x^3)*P(n-1,x) + ((n+2)*x+(2*n^2+18*n+26)*x^2+(14*n^2+66*n+56)*x^3-(16*n^2+112*n+192)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (2*n^3 + 16*n^2 + 22*n - 4)*x + (6*n^3 + 20*n^2 - 26*n - 64)*x^2 - (8*n^3 + 64*n^2 + 152*n + 96)*x^3)*P[n - 1, x] + ((n + 2)*x + (2*n^2 + 18*n + 26)*x^2 + (14*n^2 + 66*n + 56)*x^3 - (16*n^2 + 112*n + 192)*x^4)*D[P[n - 1, x], x])/(-n^2 - 3*n - 2 - (2*n^3 + 8*n^2 + 4*n - 8)*x + (2*n^3 + 18*n^2 + 52*n + 48)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 3*n + 2 + (2*n^3 + 16*n^2 + 22*n - 4)*x + (6*n^3 + 20*n^2 - 26*n - 64)*x^2 - (8*n^3 + 64*n^2 + 152*n + 96)*x^3)*v[i - 1] + ((n + 2)*x + (2*n^2 + 18*n + 26)*x^2 + (14*n^2 + 66*n + 56)*x^3 - (16*n^2 + 112*n + 192)*x^4)*deriv(v[i - 1], x))/(-n^2 - 3*n - 2 - (2*n^3 + 8*n^2 + 4*n - 8)*x + (2*n^3 + 18*n^2 + 52*n + 48)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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.