Real-rooted atlas

A152198 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(96 + 108\,x + 72\,n x + 27\,n x^{2} + 27\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-504\,x - 648\,x^{2} - 162\,x^{3} + 96\,n + 576\,n x + 216\,n x^{2} - 81\,n x^{3} + 108\,n^{2} x + 459\,n^{2} x^{2} + 162\,n^{2} x^{3} + 27\,n^{3} x^{2} + 81\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-64 - 528\,x - 1224\,x^{2} - 1080\,x^{3} - 324\,x^{4} - 96\,n x - 612\,n x^{2} - 1026\,n x^{3} - 486\,n x^{4} - 36\,n^{2} x^{2} - 162\,n^{2} x^{3} - 162\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2268
3268
44242755
54242755
6872162108350
7872162108350

All 15 rows

Technical details
Node id
n-bd565809281b0df1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5201f89336acf752

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): yes

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:  2,   6;
  4:  4,  24,  27;
  5:  4,  24,  27;
  6:  8,  72, 162, 108;
  7:  8,  72, 162, 108;
  8: 16, 192, 648, 864, 405;
  9: 16, 192, 648, 864, 405;
%F Formula
The row polynomials satisfy (96+(72*n+108)*x+(27*n^2+27*n)*x^2)*P(n,x) = (96*n+(108*n^2+576*n-504)*x+(27*n^3+459*n^2+216*n-648)*x^2+(81*n^3+162*n^2-81*n-162)*x^3)*P(n-1,x) - (64+(96*n+528)*x+(36*n^2+612*n+1224)*x^2+(162*n^2+1026*n+1080)*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[((96*n + (108*n^2 + 576*n - 504)*x + (27*n^3 + 459*n^2 + 216*n - 648)*x^2 + (81*n^3 + 162*n^2 - 81*n - 162)*x^3)*P[n - 1, x] - (64 + (96*n + 528)*x + (36*n^2 + 612*n + 1224)*x^2 + (162*n^2 + 1026*n + 1080)*x^3 + (162*n^2 + 486*n + 324)*x^4)*D[P[n - 1, x], x])/(96 + (72*n + 108)*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] = ((96*n + (108*n^2 + 576*n - 504)*x + (27*n^3 + 459*n^2 + 216*n - 648)*x^2 + (81*n^3 + 162*n^2 - 81*n - 162)*x^3)*v[i - 1] - (64 + (96*n + 528)*x + (36*n^2 + 612*n + 1224)*x^2 + (162*n^2 + 1026*n + 1080)*x^3 + (162*n^2 + 486*n + 324)*x^4)*deriv(v[i - 1], x))/(96 + (72*n + 108)*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

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.