Real-rooted atlas

A100861 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(24 + 72\,x + 36\,n x + 27\,n x^{2} + 27\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-108\,x - 432\,x^{2} - 162\,x^{3} + 24\,n + 162\,n x + 216\,n x^{2} - 81\,n x^{3} + 54\,n^{2} x + 243\,n^{2} x^{2} + 162\,n^{2} x^{3} + 27\,n^{3} x^{2} + 81\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 - 96\,x - 396\,x^{2} - 648\,x^{3} - 324\,x^{4} - 24\,n x - 198\,n x^{2} - 540\,n x^{3} - 486\,n x^{4} - 18\,n^{2} x^{2} - 108\,n^{2} x^{3} - 162\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2268
341822
4107281163
526240405671
676900243016205026
72323276113401134026188

All 15 rows

Technical details
Node id
n-304fef7143093e4a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4744a44d4b95ee5f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:    2,     6;
  3:    4,    18;
  4:   10,    72,     81;
  5:   26,   240,    405;
  6:   76,   900,   2430,   1620;
  7:  232,  3276,  11340,  11340;
  8:  764, 12768,  56700,  90720,  42525;
  9: 2620, 50112, 265356, 544320, 382725;
%F Formula
The row polynomials satisfy (24+(36*n+72)*x+(27*n^2+27*n)*x^2)*P(n,x) = (24*n+(54*n^2+162*n-108)*x+(27*n^3+243*n^2+216*n-432)*x^2+(81*n^3+162*n^2-81*n-162)*x^3)*P(n-1,x) - (8+(24*n+96)*x+(18*n^2+198*n+396)*x^2+(108*n^2+540*n+648)*x^3+(162*n^2+486*n+324)*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[((24*n + (54*n^2 + 162*n - 108)*x + (27*n^3 + 243*n^2 + 216*n - 432)*x^2 + (81*n^3 + 162*n^2 - 81*n - 162)*x^3)*P[n - 1, x] - (8 + (24*n + 96)*x + (18*n^2 + 198*n + 396)*x^2 + (108*n^2 + 540*n + 648)*x^3 + (162*n^2 + 486*n + 324)*x^4)*D[P[n - 1, x], x])/(24 + (36*n + 72)*x + (27*n^2 + 27*n)*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; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + (54*n^2 + 162*n - 108)*x + (27*n^3 + 243*n^2 + 216*n - 432)*x^2 + (81*n^3 + 162*n^2 - 81*n - 162)*x^3)*v[i - 1] - (8 + (24*n + 96)*x + (18*n^2 + 198*n + 396)*x^2 + (108*n^2 + 540*n + 648)*x^3 + (162*n^2 + 486*n + 324)*x^4)*deriv(v[i - 1], x))/(24 + (36*n + 72)*x + (27*n^2 + 27*n)*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.