Real-rooted atlas

A086645 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-10 + 100\,x + 9\,n - 180\,n x - 2\,n^{2} + 121\,n^{2} x - 36\,n^{3} x + 4\,n^{4} x\right)\,P_{n}(x) = \left(-3 - 4\,x + 67\,x^{2} + 5\,n - 31\,n x - 198\,n x^{2} - 2\,n^{2} + 59\,n^{2} x + 203\,n^{2} x^{2} - 28\,n^{3} x - 84\,n^{3} x^{2} + 4\,n^{4} x + 12\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(4\,x - 18\,x^{2} + 14\,x^{3} - 4\,n x + 42\,n x^{2} - 38\,n x^{3} - 32\,n^{2} x^{2} + 32\,n^{2} x^{3} + 8\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
36410
41560984
52828025216576
6458401890720253520
76619808316792016503619968
8914004270274804825025327649107520

All 15 rows

Technical details
Node id
n-cbb9afbd49c5dd9d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-dbbfaec00ea58ed5

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:   0;
   2:   1;
   3:   6,     4;
   4:  15,    60,      9;
   5:  28,   280,    252,     16;
   6:  45,   840,   1890,    720,      25;
   7:  66,  1980,   8316,   7920,    1650,     36;
   8:  91,  4004,  27027,  48048,   25025,   3276,     49;
   9: 120,  7280,  72072, 205920,  200200,  65520,   5880,   64;
  10: 153, 12240, 167076, 700128, 1093950, 668304, 149940, 9792, 81;
%F Formula
The row polynomials satisfy -(2*n^2-9*n+10-(4*n^4-36*n^3+121*n^2-180*n+100)*x)*P(n,x) = -(2*n^2-5*n+3-(4*n^4-28*n^3+59*n^2-31*n-4)*x-(12*n^4-84*n^3+203*n^2-198*n+67)*x^2)*P(n-1,x) - ((4*n-4)*x-(8*n^3-32*n^2+42*n-18)*x^2+(8*n^3-32*n^2+38*n-14)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 3 - (4*n^4 - 28*n^3 + 59*n^2 - 31*n - 4)*x - (12*n^4 - 84*n^3 + 203*n^2 - 198*n + 67)*x^2)*P[n - 1, x] - ((4*n - 4)*x - (8*n^3 - 32*n^2 + 42*n - 18)*x^2 + (8*n^3 - 32*n^2 + 38*n - 14)*x^3)*D[P[n - 1, x], x])/(-2*n^2 + 9*n - 10 + (4*n^4 - 36*n^3 + 121*n^2 - 180*n + 100)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 5*n + 3 - (4*n^4 - 28*n^3 + 59*n^2 - 31*n - 4)*x - (12*n^4 - 84*n^3 + 203*n^2 - 198*n + 67)*x^2)*v[i - 1] - ((4*n - 4)*x - (8*n^3 - 32*n^2 + 42*n - 18)*x^2 + (8*n^3 - 32*n^2 + 38*n - 14)*x^3)*deriv(v[i - 1], x))/(-2*n^2 + 9*n - 10 + (4*n^4 - 36*n^3 + 121*n^2 - 180*n + 100)*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

Same fingerprint ?

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