Real-rooted atlas

A122848 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,n x + n x^{2} - 4\,n x^{3} + n^{2} x^{2}\right)\,P_{n}(x) = \left(4 + 14\,x + 12\,x^{2} - 4\,x^{3} - 8\,x^{4} - 2\,n + 3\,n x + 14\,n x^{2} + 2\,n x^{3} - 8\,n x^{4} - 3\,n^{2} x + n^{2} x^{2} + 6\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-1}(x) + \left(1 + 4\,x + 2\,x^{2} - 8\,x^{3} - 8\,x^{4} + 2\,n x + 5\,n x^{2} - 2\,n x^{3} - 8\,n x^{4} + n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
22121226
34367232144
41011232432080846
5263601380208012001925238
676122457601168010800403244834020
7232431224192616007840048384125441024230688

All 15 rows

Technical details
Node id
n-2057ae96e4f2c5de · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9776d3e5efdce8d6

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 0)
  • 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;
  1:    1,     4;
  2:    2,    12,     12;
  3:    4,    36,     72,      32;
  4:   10,   112,    324,     320,      80;
  5:   26,   360,   1380,    2080,    1200,     192;
  6:   76,  1224,   5760,   11680,   10800,    4032,     448;
  7:  232,  4312,  24192,   61600,   78400,   48384,   12544,   1024;
  8:  764, 15936, 103152,  315392,  512400,  446208,  194432,  36864,   2304;
  9: 2620, 60912, 447984, 1599360, 3160080, 3568320, 2239104, 718848, 103680, 5120;
%F Formula
The row polynomials satisfy (2+(2*n+4)*x+(n^2+n)*x^2-4*n*x^3)*P(n,x) = -(2*n-4+(3*n^2-3*n-14)*x+(n^3-n^2-14*n-12)*x^2-(6*n^2+2*n-4)*x^3+(8*n+8)*x^4)*P(n-1,x) + (1+(2*n+4)*x+(n^2+5*n+2)*x^2+(2*n^2-2*n-8)*x^3-(8*n+8)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (3*n^2 - 3*n - 14)*x + (n^3 - n^2 - 14*n - 12)*x^2 - (6*n^2 + 2*n - 4)*x^3 + (8*n + 8)*x^4)*P[n - 1, x] + (1 + (2*n + 4)*x + (n^2 + 5*n + 2)*x^2 + (2*n^2 - 2*n - 8)*x^3 - (8*n + 8)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 4)*x + (n^2 + n)*x^2 - 4*n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 4 + (3*n^2 - 3*n - 14)*x + (n^3 - n^2 - 14*n - 12)*x^2 - (6*n^2 + 2*n - 4)*x^3 + (8*n + 8)*x^4)*v[i - 1] + (1 + (2*n + 4)*x + (n^2 + 5*n + 2)*x^2 + (2*n^2 - 2*n - 8)*x^3 - (8*n + 8)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 4)*x + (n^2 + n)*x^2 - 4*n*x^3)); 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.