Real-rooted atlas

A091044 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + x - 3\,n - 6\,n x + 2\,n^{2} + 13\,n^{2} x - 12\,n^{3} x + 4\,n^{4} x\right)\,P_{n}(x) = \left(1 - x + 4\,x^{2} + 3\,n - 14\,n x + 3\,n x^{2} + 2\,n^{2} + 7\,n^{2} x - 5\,n^{2} x^{2} - 16\,n^{3} x^{2} + 4\,n^{4} x + 12\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x^{2} + 2\,x^{3} + 4\,n x - 6\,n x^{2} + 2\,n x^{3} + 8\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
2101222
32811236176
460504540801184
51101584356417601507168
618240041544416016455025240448
7280873651480915205460010080392217088

All 15 rows

Technical details
Node id
n-e4c987ffe78da50b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b4e36ba3efa876e4

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 -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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   2;
  2:  10,    12;
  3:  28,   112,     36;
  4:  60,   504,    540,      80;
  5: 110,  1584,   3564,    1760,     150;
  6: 182,  4004,  15444,   16016,    4550,     252;
  7: 280,  8736,  51480,   91520,   54600,   10080,    392;
  8: 408, 17136, 143208,  388960,  397800,  154224,  19992,   576;
  9: 570, 31008, 348840, 1343680, 2099500, 1395360, 379848, 36480, 810;
%F Formula
The row polynomials satisfy (2*n^2-3*n+1+(4*n^4-12*n^3+13*n^2-6*n+1)*x)*P(n,x) = (2*n^2+3*n+1+(4*n^4+7*n^2-14*n-1)*x+(12*n^4-16*n^3-5*n^2+3*n+4)*x^2)*P(n-1,x) + (4*n*x+(8*n^3-6*n-2)*x^2-(8*n^3-2*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 3*n + 1 + (4*n^4 + 7*n^2 - 14*n - 1)*x + (12*n^4 - 16*n^3 - 5*n^2 + 3*n + 4)*x^2)*P[n - 1, x] + (4*n*x + (8*n^3 - 6*n - 2)*x^2 - (8*n^3 - 2*n - 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1 + (4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 3*n + 1 + (4*n^4 + 7*n^2 - 14*n - 1)*x + (12*n^4 - 16*n^3 - 5*n^2 + 3*n + 4)*x^2)*v[i - 1] + (4*n*x + (8*n^3 - 6*n - 2)*x^2 - (8*n^3 - 2*n - 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1 + (4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)*x)); 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

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