Real-rooted atlas

A086645 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12 - 18\,x - 14\,n + 39\,n x + 4\,n^{2} - 27\,n^{2} x + 6\,n^{3} x\right)\,P_{n}(x) = \left(48 + 144\,x + 36\,x^{2} - 56\,n - 96\,n x + 108\,n x^{2} + 16\,n^{2} - 36\,n^{2} x - 171\,n^{2} x^{2} + 24\,n^{3} x + 54\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(12\,x + 42\,x^{2} + 18\,x^{3} - 8\,n x - 12\,n x^{2} + 36\,n x^{3} - 12\,n^{2} x^{2} - 36\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2268
38482783
432288486108914
51281536432034564059845
651276803024043200202501458103340
720483686418662438707234020010497651031062887
8819217203210644482903040381024022861445000941749610761686

All 15 rows

Technical details
Node id
n-cc3556722af3bd66 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-98685e9cd7383975

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:      2,       6;
   3:      8,      48,       27;
   4:     32,     288,      486,       108;
   5:    128,    1536,     4320,      3456,       405;
   6:    512,    7680,    30240,     43200,     20250,      1458;
   7:   2048,   36864,   186624,    387072,    340200,    104976,      5103;
   8:   8192,  172032,  1064448,   2903040,   3810240,   2286144,    500094,    17496;
   9:  32768,  786432,  5750784,  19464192,  34214400,  31352832,  13716864,  2239488,   59049;
  10: 131072, 3538944, 29859840, 120766464, 266872320, 332563968, 226328256, 75582720, 9565938, 196830;
%F Formula
The row polynomials satisfy (4*n^2-14*n+12+(6*n^3-27*n^2+39*n-18)*x)*P(n,x) = (16*n^2-56*n+48+(24*n^3-36*n^2-96*n+144)*x+(54*n^3-171*n^2+108*n+36)*x^2)*P(n-1,x) - ((8*n-12)*x+(12*n^2+12*n-42)*x^2+(36*n^2-36*n-18)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 6*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 56*n + 48 + (24*n^3 - 36*n^2 - 96*n + 144)*x + (54*n^3 - 171*n^2 + 108*n + 36)*x^2)*P[n - 1, x] - ((8*n - 12)*x + (12*n^2 + 12*n - 42)*x^2 + (36*n^2 - 36*n - 18)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12 + (6*n^3 - 27*n^2 + 39*n - 18)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 6*x; for(n = 3, N, my(i = n); v[i] = ((16*n^2 - 56*n + 48 + (24*n^3 - 36*n^2 - 96*n + 144)*x + (54*n^3 - 171*n^2 + 108*n + 36)*x^2)*v[i - 1] - ((8*n - 12)*x + (12*n^2 + 12*n - 42)*x^2 + (36*n^2 - 36*n - 18)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12 + (6*n^3 - 27*n^2 + 39*n - 18)*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