Real-rooted atlas

A035607 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4 - 18\,x - 23\,x^{2} - 4\,x^{3} + 4\,x^{4} + 2\,n x - 2\,n x^{2} - 10\,n x^{3} - 6\,n x^{4} + 4\,n^{2} x^{2} + 2\,n^{2} x^{3} + 2\,n^{2} x^{4} + 2\,n^{3} x^{3}\right)\,P_{n}(x) = \left(4 + 18\,x + 23\,x^{2} + 4\,x^{3} - 4\,x^{4} - 8\,n - 22\,n x - 12\,n x^{2} + 6\,n x^{3} + 6\,n x^{4} - 12\,n^{2} x - 28\,n^{2} x^{2} - 14\,n^{2} x^{3} - 2\,n^{2} x^{4} - 4\,n^{3} x^{2} - 6\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(4 + 28\,x + 74\,x^{2} + 90\,x^{3} + 48\,x^{4} + 8\,x^{5} + 8\,n x + 44\,n x^{2} + 88\,n x^{3} + 78\,n x^{4} + 30\,n x^{5} + 4\,n x^{6} + 4\,n^{2} x^{2} + 16\,n^{2} x^{3} + 18\,n^{2} x^{4} + 4\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
23811
37322463
41711216864361
5413527926401601985
699104831443904208038410659
72393008112801894415680614489656191

All 15 rows

Technical details
Node id
n-32b958a03883d806 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-920ef5a99590eda1

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  2:    3,     8;
  3:    7,    32,     24;
  4:   17,   112,    168,     64;
  5:   41,   352,    792,    640,    160;
  6:   99,  1048,   3144,   3904,   2080,    384;
  7:  239,  3008,  11280,  18944,  15680,   6144,    896;
  8:  577,  8416,  37872,  80256,  90560,  55296,  17024,  2048;
  9: 1393, 23104, 121296, 310528, 444800, 371712, 178304, 45056, 4608;
%F Formula
The row polynomials satisfy -(4-(2*n-18)*x-(4*n^2-2*n-23)*x^2-(2*n^3+2*n^2-10*n-4)*x^3-(2*n^2-6*n+4)*x^4)*P(n,x) = -(8*n-4+(12*n^2+22*n-18)*x+(4*n^3+28*n^2+12*n-23)*x^2+(6*n^3+14*n^2-6*n-4)*x^3+(2*n^2-6*n+4)*x^4)*P(n-1,x) + (4+(8*n+28)*x+(4*n^2+44*n+74)*x^2+(16*n^2+88*n+90)*x^3+(18*n^2+78*n+48)*x^4+(4*n^2+30*n+8)*x^5+4*n*x^6)*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[(-(8*n - 4 + (12*n^2 + 22*n - 18)*x + (4*n^3 + 28*n^2 + 12*n - 23)*x^2 + (6*n^3 + 14*n^2 - 6*n - 4)*x^3 + (2*n^2 - 6*n + 4)*x^4)*P[n - 1, x] + (4 + (8*n + 28)*x + (4*n^2 + 44*n + 74)*x^2 + (16*n^2 + 88*n + 90)*x^3 + (18*n^2 + 78*n + 48)*x^4 + (4*n^2 + 30*n + 8)*x^5 + 4*n*x^6)*D[P[n - 1, x], x])/(-4 + (2*n - 18)*x + (4*n^2 - 2*n - 23)*x^2 + (2*n^3 + 2*n^2 - 10*n - 4)*x^3 + (2*n^2 - 6*n + 4)*x^4)]];
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] = (-(8*n - 4 + (12*n^2 + 22*n - 18)*x + (4*n^3 + 28*n^2 + 12*n - 23)*x^2 + (6*n^3 + 14*n^2 - 6*n - 4)*x^3 + (2*n^2 - 6*n + 4)*x^4)*v[i - 1] + (4 + (8*n + 28)*x + (4*n^2 + 44*n + 74)*x^2 + (16*n^2 + 88*n + 90)*x^3 + (18*n^2 + 78*n + 48)*x^4 + (4*n^2 + 30*n + 8)*x^5 + 4*n*x^6)*deriv(v[i - 1], x))/(-4 + (2*n - 18)*x + (4*n^2 - 2*n - 23)*x^2 + (2*n^3 + 2*n^2 - 10*n - 4)*x^3 + (2*n^2 - 6*n + 4)*x^4)); 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.