Real-rooted atlas

A207607 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-28 - 352\,x + 832\,x^{2} + 1024\,x^{3} + 728\,n + 2304\,n x - 640\,n x^{2} - 1024\,n x^{3} - 960\,n^{2} - 3072\,n^{2} x + 128\,n^{3} + 1024\,n^{3} x\right)\,P_{n}(x) = \left(512 + 3600\,x + 7168\,x^{2} + 6400\,x^{3} + 2048\,x^{4} - 812\,n - 3408\,n x - 5824\,n x^{2} - 4864\,n x^{3} - 2048\,n x^{4} - 1344\,n^{2} - 4096\,n^{2} x - 2048\,n^{2} x^{2} + 192\,n^{3} + 1792\,n^{3} x + 2048\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-505 - 4024\,x - 10976\,x^{2} - 13824\,x^{3} - 8448\,x^{4} - 2048\,x^{5} - 320\,n - 2176\,n x - 4096\,n x^{2} - 2048\,n x^{3} + 80\,n^{2} + 1024\,n^{2} x + 3328\,n^{2} x^{2} + 2048\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111819
24410032176
31536126241281517
449928765248339251212527
51564117883148836416171522048100456
6476944424157104266880224256829448192788569
714251158128697584157542419443201278976389120327686090571

All 15 rows

Technical details
Node id
n-afbd6e41feca69f1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ec5faffac26d6c0b

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:     11,       8;
  2:     44,     100,       32;
  3:    153,     612,      624,      128;
  4:    499,    2876,     5248,     3392,      512;
  5:   1564,   11788,    31488,    36416,    17152,     2048;
  6:   4769,   44424,   157104,   266880,   224256,    82944,     8192;
  7:  14251,  158128,   697584,  1575424,  1944320,  1278976,   389120,    32768;
  8:  41932,  540020,  2857888,  8083264, 13277440, 12813312,  6914048,  1785856,  131072;
  9: 121881, 1787052, 11037504, 37582656, 77435136, 99449856, 78618624, 35930112, 8060928, 524288;
%F Formula
The row polynomials satisfy (128*n^3-960*n^2+728*n-28+(1024*n^3-3072*n^2+2304*n-352)*x-(640*n-832)*x^2-(1024*n-1024)*x^3)*P(n,x) = (192*n^3-1344*n^2-812*n+512+(1792*n^3-4096*n^2-3408*n+3600)*x+(2048*n^3-2048*n^2-5824*n+7168)*x^2-(4864*n-6400)*x^3-(2048*n-2048)*x^4)*P(n-1,x) + (80*n^2-320*n-505+(1024*n^2-2176*n-4024)*x+(3328*n^2-4096*n-10976)*x^2+(2048*n^2-2048*n-13824)*x^3-8448*x^4-2048*x^5)*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[((192*n^3 - 1344*n^2 - 812*n + 512 + (1792*n^3 - 4096*n^2 - 3408*n + 3600)*x + (2048*n^3 - 2048*n^2 - 5824*n + 7168)*x^2 - (4864*n - 6400)*x^3 - (2048*n - 2048)*x^4)*P[n - 1, x] + (80*n^2 - 320*n - 505 + (1024*n^2 - 2176*n - 4024)*x + (3328*n^2 - 4096*n - 10976)*x^2 + (2048*n^2 - 2048*n - 13824)*x^3 - 8448*x^4 - 2048*x^5)*D[P[n - 1, x], x])/(128*n^3 - 960*n^2 + 728*n - 28 + (1024*n^3 - 3072*n^2 + 2304*n - 352)*x - (640*n - 832)*x^2 - (1024*n - 1024)*x^3)]];
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] = ((192*n^3 - 1344*n^2 - 812*n + 512 + (1792*n^3 - 4096*n^2 - 3408*n + 3600)*x + (2048*n^3 - 2048*n^2 - 5824*n + 7168)*x^2 - (4864*n - 6400)*x^3 - (2048*n - 2048)*x^4)*v[i - 1] + (80*n^2 - 320*n - 505 + (1024*n^2 - 2176*n - 4024)*x + (3328*n^2 - 4096*n - 10976)*x^2 + (2048*n^2 - 2048*n - 13824)*x^3 - 8448*x^4 - 2048*x^5)*deriv(v[i - 1], x))/(128*n^3 - 960*n^2 + 728*n - 28 + (1024*n^3 - 3072*n^2 + 2304*n - 352)*x - (640*n - 832)*x^2 - (1024*n - 1024)*x^3)); 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