Real-rooted atlas

A114656 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 2\,n + 4\,n x - 4\,n x^{2} + 4\,n^{2} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(2 + 16\,x + 16\,x^{3} - 10\,n - 16\,n x - 44\,n x^{2} - 24\,n x^{3} + 16\,n^{2} x + 24\,n^{2} x^{2} - 16\,n^{2} x^{3} + 20\,n^{3} x^{2} + 24\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 + 2\,x^{2} + 8\,x^{4} + 4\,n x + 2\,n x^{2} - 16\,n x^{3} - 8\,n x^{4} + 4\,n^{2} x^{2} - 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
311321255
44520418032461
519712161752768804013
690370601428010560280019235795
7427940512106020112640516009216448324715
82079323119674415610404806972002190722822410242982145

All 15 rows

Technical details
Node id
n-d4a188d61f4db1a1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-cfbd64f549423676

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:      3,       4;
   3:     11,      32,       12;
   4:     45,     204,      180,       32;
   5:    197,    1216,     1752,      768,       80;
   6:    903,    7060,    14280,    10560,     2800,      192;
   7:   4279,   40512,   106020,   112640,    51600,     9216,      448;
   8:  20793,  231196,   744156,  1040480,   697200,   219072,    28224,    1024;
   9: 103049, 1316096,  5034288,  8770048,  7841120,  3655680,   840448,   81920,   2304;
  10: 518859, 7484580, 33192720, 69417600, 78170400, 49182336, 16972032, 2985984, 228096, 5120;
%F Formula
The row polynomials satisfy -(2*n+2-(4*n^2+4*n)*x-(4*n^3-4*n)*x^2)*P(n,x) = -(10*n-2-(16*n^2-16*n+16)*x-(20*n^3+24*n^2-44*n)*x^2-(24*n^3-16*n^2-24*n+16)*x^3)*P(n-1,x) - (1-4*n*x-(4*n^2+2*n+2)*x^2+16*n*x^3+(16*n^2+8*n-8)*x^4)*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[(-(10*n - 2 - (16*n^2 - 16*n + 16)*x - (20*n^3 + 24*n^2 - 44*n)*x^2 - (24*n^3 - 16*n^2 - 24*n + 16)*x^3)*P[n - 1, x] - (1 - 4*n*x - (4*n^2 + 2*n + 2)*x^2 + 16*n*x^3 + (16*n^2 + 8*n - 8)*x^4)*D[P[n - 1, x], x])/(-2*n - 2 + (4*n^2 + 4*n)*x + (4*n^3 - 4*n)*x^2)]];
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] = (-(10*n - 2 - (16*n^2 - 16*n + 16)*x - (20*n^3 + 24*n^2 - 44*n)*x^2 - (24*n^3 - 16*n^2 - 24*n + 16)*x^3)*v[i - 1] - (1 - 4*n*x - (4*n^2 + 2*n + 2)*x^2 + 16*n*x^3 + (16*n^2 + 8*n - 8)*x^4)*deriv(v[i - 1], x))/(-2*n - 2 + (4*n^2 + 4*n)*x + (4*n^3 - 4*n)*x^2)); 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