Real-rooted atlas

A116414 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(10 + 2\,x + 5\,n + n x\right)\,P_{n-1}(x) + \left(13 + 10\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
210212
369303102
4410282604756
52252214071410055211
6117901435866001440150634344
75976788950523951570025352107219564
829605052121037560014404031850407428081373112
9144115529313002502630117390033169557960613236098445141
1069209001597518015783300877422030303006756129744087844501051266196
113287679284923800953713806139380025124190684054012570181542001210555011307954386
121547902204424470445568927004078081801932723006232489213973400218124023265016170660121833939468
137233782532267044260316201992625973831001400749155523029600138808320264132003580830337700210547801310842766191
143359301370114546366741753998792015979517528967065775041090723101262322600284160360469210505618690474760268329101463712698768
Technical details
Node id
n-f4f552e83b5e543f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b523dc1f3adb9012

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:       1;
  2:      10,       2;
  3:      69,      30,       3;
  4:     410,     282,      60,       4;
  5:    2252,    2140,     714,     100,      5;
  6:   11790,   14358,    6600,    1440,    150,     6;
  7:   59767,   88950,   52395,   15700,   2535,   210,    7;
  8:  296050,  521210,  375600,  144040,  31850,  4074,  280,   8;
  9: 1441155, 2931300, 2502630, 1173900, 331695, 57960, 6132, 360, 9;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (5*n+10+(n+2)*x)*P(n-1,x) + (13+10*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 10 + (n + 2)*x)*P[n - 1, x] + (13 + 10*x + x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 10 + (n + 2)*x)*v[i - 1] + (13 + 10*x + x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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