Real-rooted atlas

A100861 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 16\,x + 8\,n x + 8\,n x^{2} + 8\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-24\,x - 128\,x^{2} - 64\,x^{3} + 4\,n + 36\,n x + 64\,n x^{2} - 32\,n x^{3} + 12\,n^{2} x + 72\,n^{2} x^{2} + 64\,n^{2} x^{3} + 8\,n^{3} x^{2} + 32\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 16\,x - 88\,x^{2} - 192\,x^{3} - 128\,x^{4} - 4\,n x - 44\,n x^{2} - 160\,n x^{3} - 192\,n x^{4} - 4\,n^{2} x^{2} - 32\,n^{2} x^{3} - 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
22810
342428
41096144250
5263207201066
6761200432038409436
72324368201602688051640

All 15 rows

Technical details
Node id
n-e9930db7b976d5ff · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-82e87d4c95b4bd9c

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,     8;
  3:    4,    24;
  4:   10,    96,    144;
  5:   26,   320,    720;
  6:   76,  1200,   4320,    3840;
  7:  232,  4368,  20160,   26880;
  8:  764, 17024, 100800,  215040,  134400;
  9: 2620, 66816, 471744, 1290240, 1209600;
%F Formula
The row polynomials satisfy (4+(8*n+16)*x+(8*n^2+8*n)*x^2)*P(n,x) = (4*n+(12*n^2+36*n-24)*x+(8*n^3+72*n^2+64*n-128)*x^2+(32*n^3+64*n^2-32*n-64)*x^3)*P(n-1,x) - (1+(4*n+16)*x+(4*n^2+44*n+88)*x^2+(32*n^2+160*n+192)*x^3+(64*n^2+192*n+128)*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[((4*n + (12*n^2 + 36*n - 24)*x + (8*n^3 + 72*n^2 + 64*n - 128)*x^2 + (32*n^3 + 64*n^2 - 32*n - 64)*x^3)*P[n - 1, x] - (1 + (4*n + 16)*x + (4*n^2 + 44*n + 88)*x^2 + (32*n^2 + 160*n + 192)*x^3 + (64*n^2 + 192*n + 128)*x^4)*D[P[n - 1, x], x])/(4 + (8*n + 16)*x + (8*n^2 + 8*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] = ((4*n + (12*n^2 + 36*n - 24)*x + (8*n^3 + 72*n^2 + 64*n - 128)*x^2 + (32*n^3 + 64*n^2 - 32*n - 64)*x^3)*v[i - 1] - (1 + (4*n + 16)*x + (4*n^2 + 44*n + 88)*x^2 + (32*n^2 + 160*n + 192)*x^3 + (64*n^2 + 192*n + 128)*x^4)*deriv(v[i - 1], x))/(4 + (8*n + 16)*x + (8*n^2 + 8*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.