Real-rooted atlas

A091044 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(1 - 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-3\,x - 4\,n - 21\,n x + 8\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,n x - 12\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
24854102
32887203241332
4153660486480162015684
576804147272576453607290174378
636864253440622080653184272160306181868346
71720321437696456192068428804898880146966412247219505544
87864327741440301916165930496061585920323326087348320472392199763688
93538944401080321857945604528742406227020804803701761939956483464208017714702015797230
1015728640201719808108291686431585075205434490880560431872033625912321080832896155889360649539020103491310
116920601699090432060515942402057542041642639851520554318069764483454976021616657920567437270467552056023383404198583267836
12301989888477521510432699842560127083479040308631306240486094307328498886262784326651719680129699947520283718635202837186352829047961946116024812
13130862284822649241600171907743744752096378880209687740416038887544586244860943073280406235956838422048991078407349663692801361849448961160667144029016678618944843751762
14563714457610599845068888332042240042976935936001353773481984028936908177408427762990448644374848765952030467696762880139643610163203968818394112631402926336464266857601004423490183371789521794
Technical details
Node id
n-6000305df5c32760 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-87080cd34b7401f9

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:       6;
  2:      48,       54;
  3:     288,      720,       324;
  4:    1536,     6048,      6480,      1620;
  5:    7680,    41472,     72576,     45360,      7290;
  6:   36864,   253440,    622080,    653184,    272160,     30618;
  7:  172032,  1437696,   4561920,   6842880,   4898880,   1469664,    122472;
  8:  786432,  7741440,  30191616,  59304960,  61585920,  32332608,   7348320,   472392;
  9: 3538944, 40108032, 185794560, 452874240, 622702080, 480370176, 193995648, 34642080, 1771470;
%F Formula
The row polynomials satisfy (2*n^2-3*n+1)*P(n,x) = (8*n^2-4*n+(18*n^2-21*n-3)*x)*P(n-1,x) - (4*n*x+12*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 4*n + (18*n^2 - 21*n - 3)*x)*P[n - 1, x] - (4*n*x + 12*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1)]];
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] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 4*n + (18*n^2 - 21*n - 3)*x)*v[i - 1] - (4*n*x + 12*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1)); 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