Real-rooted atlas

A091042 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 3\,x - 10\,n + 11\,n x + 4\,n^{2} - 12\,n^{2} x + 4\,n^{3} x\right)\,P_{n}(x) = \left(24 + 60\,x + 24\,x^{2} - 40\,n - 76\,n x - 10\,n x^{2} + 16\,n^{2} - 40\,n^{2} x^{2} + 16\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(8\,x + 22\,x^{2} + 12\,x^{3} - 8\,n x - 16\,n x^{2} - 8\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
241216
3168060156
4644486722241408
525623045184384072012304
61024112643379239424176002112105216
7409653248199680319488232960698885824885184
8163842457601105920225280023040001161216250880153607352320

All 15 rows

Technical details
Node id
n-4220c6692cf7fad9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4a484cf9cab203b5

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,      12;
   3:     16,      80,       60;
   4:     64,     448,      672,      224;
   5:    256,    2304,     5184,     3840,       720;
   6:   1024,   11264,    33792,    39424,     17600,      2112;
   7:   4096,   53248,   199680,   319488,    232960,     69888,     5824;
   8:  16384,  245760,  1105920,  2252800,   2304000,   1161216,   250880,    15360;
   9:  65536, 1114112,  5849088, 14483456,  19148800,  13787136,  5117952,   835584,   39168;
  10: 262144, 4980736, 29884416, 87162880, 141639680, 133545984, 71909376, 20545536, 2626560, 97280;
%F Formula
The row polynomials satisfy (4*n^2-10*n+6+(4*n^3-12*n^2+11*n-3)*x)*P(n,x) = (16*n^2-40*n+24+(16*n^3-76*n+60)*x+(24*n^3-40*n^2-10*n+24)*x^2)*P(n-1,x) - ((8*n-8)*x+(8*n^2+16*n-22)*x^2+(16*n^2-12)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 40*n + 24 + (16*n^3 - 76*n + 60)*x + (24*n^3 - 40*n^2 - 10*n + 24)*x^2)*P[n - 1, x] - ((8*n - 8)*x + (8*n^2 + 16*n - 22)*x^2 + (16*n^2 - 12)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6 + (4*n^3 - 12*n^2 + 11*n - 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n^2 - 40*n + 24 + (16*n^3 - 76*n + 60)*x + (24*n^3 - 40*n^2 - 10*n + 24)*x^2)*v[i - 1] - ((8*n - 8)*x + (8*n^2 + 16*n - 22)*x^2 + (16*n^2 - 12)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6 + (4*n^3 - 12*n^2 + 11*n - 3)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023