Real-rooted atlas

A358735 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(n - n^{2}\right)\,P_{n-4}(x) + \left(-2\,n + 4\,n x - 2\,n^{2} - 6\,n^{2} x + 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-3}(x) + \left(-2 + 6\,n - 4\,n x + 2\,n x^{2} + n^{2} + 2\,n^{2} x - 5\,n^{2} x^{2} + 4\,n^{3} x + 4\,n^{3} x^{2} - n^{4} - 2\,n^{4} x - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(4 - 6\,n + 4\,n x - 2\,n^{2} - 6\,n^{2} x + 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 8 + 4\,x\)
\(P_{3}(x) = 60 + 68\,x + 18\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
28412
3606818146
4484884516961980
542831096010224411260030179
641604138192179916114592356404320514264
74418401815224306003627064321322840338256352809719908
8510371625083488522031325959011240257080160775043509856322560202147448
963784717365970032909801504127922238411118938006112907522073869283964953632659204592265573
108582292285642251236163549653962742176521629292748320206621371209616858720284560819248524832036288000113216099748
1112375808436918643390523048018940385950765512327567431060806546607696323899758694721578509094404151936361664025568004390848003011710252598
1219047537631215774778979565904429654072131804802566401949125900528020041689365280146152960887367555307619584271177311974464319102208090660936960574801920086007788361844
1331173353801032852266391440011902246192718429945557463561650579494614352060403432019193652280508750164833031330527411215113145136896048811518948480105571065853441371957212160809512704002625018680353863
145406903054002454207053537532824974309033832726999861336553984133137004354143201814186252891683218216160000405920136423389767925767617826006850880313127480011520092052043623513618318137704243222103238044160122049607680085283620101746864
Technical details
Node id
n-5433b0a58dea9df1 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8e500248e9eaf559

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:        8,         4;
  3:       60,        68,        18;
  4:      484,       884,       516,         96;
  5:     4283,     10960,     10224,       4112,        600;
  6:    41604,    138192,    179916,     114592,      35640,      4320;
  7:   441840,   1815224,   3060036,    2706432,    1322840,    338256,     35280;
  8:  5103716,  25083488,  52203132,   59590112,   40257080,  16077504,   3509856,   322560;
  9: 63784717, 365970032, 909801504, 1279222384, 1111893800, 611290752, 207386928, 39649536, 3265920;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (2*n^3-2*n^2-2*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (n^4-n^2-6*n+2+(2*n^4-4*n^3-2*n^2+4*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (2*n^3-2*n^2-6*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 4*x, P(3,x) = 60 + 68*x + 18*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 8 + 4*x;
P[3, x_] := 60 + 68*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (2*n^3 - 2*n^2 - 6*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*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; v[3] = 8 + 4*x; v[4] = 60 + 68*x + 18*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (2*n^3 - 2*n^2 - 6*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*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