Real-rooted atlas

A086645 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-10 + 20\,x - 4\,x^{2} + 19\,n - 18\,n x + 2\,n x^{2} - 11\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(60\,x + 80\,x^{2} - 24\,x^{3} + 40\,n + 6\,n x - 88\,n x^{2} + 12\,n x^{3} - 36\,n^{2} - 42\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(12\,x^{2} + 20\,x^{3} - 8\,x^{4} + 8\,n x + 8\,n x^{2} - 16\,n x^{3} - 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
32424452
4128240968472
564017921408320164176
63072115201408064009603236064
714336675841136648243224960268864305728
8655363727368028168171524014088780871681282554752
9294912196608051773446848512479232017203202867201843225621104896
1013107201002700831260672511180804703846424330240670924888473646080512172725760
1157671684980736017956864035061760040255488027646361611083776024330240261120011264010241402672128
12251658242422210569919528962254962688311427072026896629761443233792463896576832634887434240270336204811316336640
1310905190411576279045309988864137866772482230006579223316135936158397890566860046336181341388827177779220545536638976409690785763328
1446976204854525952002769918361680916512768150218997760184924766208152945295360843516477443021419315267007283208528199685537792014909448192724803379200
15201326592025367150592141398376448459192401920963070918656136671395840013371483095049022702878724136475033601249053245442361655296025882787841461125123440640163845762081898496
Technical details
Node id
n-730ae2ffeabcee67 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2865814b75bf33d6

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       4,        2;
   3:      24,       24,        4;
   4:     128,      240,       96,        8;
   5:     640,     1792,     1408,      320,       16;
   6:    3072,    11520,    14080,     6400,      960,       32;
   7:   14336,    67584,   113664,    82432,    24960,     2688,      64;
   8:   65536,   372736,   802816,   817152,   401408,    87808,    7168,    128;
   9:  294912,  1966080,  5177344,  6848512,  4792320,  1720320,  286720,  18432,   256;
  10: 1310720, 10027008, 31260672, 51118080, 47038464, 24330240, 6709248, 884736, 46080, 512;
%F Formula
The row polynomials satisfy (2*n^3-11*n^2+19*n-10+(4*n^2-18*n+20)*x+(2*n-4)*x^2)*P(n,x) = (8*n^3-36*n^2+40*n+(12*n^3-42*n^2+6*n+60)*x+(24*n^2-88*n+80)*x^2+(12*n-24)*x^3)*P(n-1,x) - ((4*n^2-8*n)*x+(8*n^2-8*n-12)*x^2+(16*n-20)*x^3+8*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 36*n^2 + 40*n + (12*n^3 - 42*n^2 + 6*n + 60)*x + (24*n^2 - 88*n + 80)*x^2 + (12*n - 24)*x^3)*P[n - 1, x] - ((4*n^2 - 8*n)*x + (8*n^2 - 8*n - 12)*x^2 + (16*n - 20)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 11*n^2 + 19*n - 10 + (4*n^2 - 18*n + 20)*x + (2*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = ((8*n^3 - 36*n^2 + 40*n + (12*n^3 - 42*n^2 + 6*n + 60)*x + (24*n^2 - 88*n + 80)*x^2 + (12*n - 24)*x^3)*v[i - 1] - ((4*n^2 - 8*n)*x + (8*n^2 - 8*n - 12)*x^2 + (16*n - 20)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 11*n^2 + 19*n - 10 + (4*n^2 - 18*n + 20)*x + (2*n - 4)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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