Real-rooted atlas

A123519 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(3 + 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(21 + 36\,x + 19\,n + 30\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(4 + 16\,x + 12\,x^{2} + 2\,n + 8\,n x + 6\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 42 + 72\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
04272114
12287926481668
2111658329720518421852
3515435928907209849638880269178
42292619972867651211145608942402799363187902
599336103809644128809771840118972807558272195955236737256
6422136514468826317872732758401199836801156135686074611213436928414940824
7176729424606288147160800493874496101760624013187784961050319872470292480906992644615105230

All 15 rows

Technical details
Node id
n-ef95c7f426a9cafe · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-20d5dcabf3a386bd

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:       42,        72;
  1:      228,       792,        648;
  2:     1116,      5832,       9720,        5184;
  3:     5154,     35928,      90720,       98496,       38880;
  4:    22926,    199728,     676512,     1114560,      894240,       279936;
  5:    99336,   1038096,    4412880,     9771840,    11897280,      7558272,      1959552;
  6:   422136,   5144688,   26317872,    73275840,   119983680,    115613568,     60746112,     13436928;
  7:  1767294,  24606288,  147160800,   493874496,  1017606240,   1318778496,   1050319872,    470292480,    90699264;
  8:  7311666, 114487992,  783781920,  3079508544,  7654500000,  12486265344,  13371982848,   9069926400,  3537271296,   604661760;
  9: 29959788, 521085672, 4018358808, 18094415040, 52696707840, 103738682880, 139841468928, 127503009792, 75280389120, 26000455680, 3990767616;
%F Formula
The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = (4*n^2+19*n+21+(6*n^2+30*n+36)*x)*P(n-1,x) + (2*n+4+(8*n+16)*x+(6*n+12)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 42 + 72*x.
%t Mathematica
P[0, x_] := 42 + 72*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 19*n + 21 + (6*n^2 + 30*n + 36)*x)*P[n - 1, x] + (2*n + 4 + (8*n + 16)*x + (6*n + 12)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 42 + 72*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 19*n + 21 + (6*n^2 + 30*n + 36)*x)*v[i - 1] + (2*n + 4 + (8*n + 16)*x + (6*n + 12)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 5*n + 3)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023