Real-rooted atlas

A207823 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(184\,n + 64\,n x - 128\,n x^{2} + 128\,n^{3}\right)\,P_{n}(x) = \left(444 + 144\,x - 448\,x^{2} - 256\,x^{3} + 1084\,n + 656\,n x - 448\,n x^{2} - 256\,n x^{3} + 960\,n^{2} + 768\,n^{2} x + 320\,n^{3} + 256\,n^{3} x\right)\,P_{n-1}(x) + \left(-21 - 544\,x - 1312\,x^{2} - 1024\,x^{3} - 256\,x^{4} + 336\,n + 640\,n x + 256\,n x^{2} + 336\,n^{2} + 640\,n^{2} x + 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19413
2647216152
3407772432641675
4243165846192230425617767
513952493926627241344115201024183504
6778893414325988965324802483205529640961858409
7426089223028448463205631040373888013916162580481638418538661

All 15 rows

Technical details
Node id
n-8c4f40de5f6f2604 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-36bdef187abcb42b

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 0)
  • 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:        1;
  1:        9,        4;
  2:       64,       72,        16;
  3:      407,      772,       432,        64;
  4:     2431,     6584,      6192,      2304,       256;
  5:    13952,    49392,     66272,     41344,     11520,      1024;
  6:    77889,   341432,    598896,    532480,    248320,     55296,      4096;
  7:   426089,  2230284,   4846320,   5631040,   3738880,   1391616,    258048,    16384;
  8:  2294976, 13976280,  36283440,  52226560,  45296640,  23984128,   7426048,  1179648,   65536;
  9: 12210295, 84849420, 256419360, 441032320, 473777920, 327527424, 144162816, 38207488, 5308416, 262144;
%F Formula
The row polynomials satisfy (128*n^3+184*n+64*n*x-128*n*x^2)*P(n,x) = (320*n^3+960*n^2+1084*n+444+(256*n^3+768*n^2+656*n+144)*x-(448*n+448)*x^2-(256*n+256)*x^3)*P(n-1,x) + (336*n^2+336*n-21+(640*n^2+640*n-544)*x+(256*n^2+256*n-1312)*x^2-1024*x^3-256*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((320*n^3 + 960*n^2 + 1084*n + 444 + (256*n^3 + 768*n^2 + 656*n + 144)*x - (448*n + 448)*x^2 - (256*n + 256)*x^3)*P[n - 1, x] + (336*n^2 + 336*n - 21 + (640*n^2 + 640*n - 544)*x + (256*n^2 + 256*n - 1312)*x^2 - 1024*x^3 - 256*x^4)*D[P[n - 1, x], x])/(128*n^3 + 184*n + 64*n*x - 128*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((320*n^3 + 960*n^2 + 1084*n + 444 + (256*n^3 + 768*n^2 + 656*n + 144)*x - (448*n + 448)*x^2 - (256*n + 256)*x^3)*v[i - 1] + (336*n^2 + 336*n - 21 + (640*n^2 + 640*n - 544)*x + (256*n^2 + 256*n - 1312)*x^2 - 1024*x^3 - 256*x^4)*deriv(v[i - 1], x))/(128*n^3 + 184*n + 64*n*x - 128*n*x^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