Real-rooted atlas

A110441 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n + 2\,n x - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(8 + 6\,x - 4\,x^{2} - 4\,x^{3} + 16\,n + 14\,n x - 4\,n x^{2} - 4\,n x^{3} + 12\,n^{2} + 12\,n^{2} x + 4\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(1 + 2\,x - 8\,x^{2} - 12\,x^{3} - 4\,x^{4} + 2\,n + 8\,n x + 4\,n x^{2} + 2\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
2466416126
3224560384641232
499637124512204825611524
54192210403788830208102401024104592
617016108032261504307200181760491524096928760
7673285185281588224250112021708801019904229376163848111744

All 15 rows

Technical details
Node id
n-4cd6b1ad9e993dce · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8c22c7448247a195

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:       8,        4;
  2:      46,       64,       16;
  3:     224,      560,      384,        64;
  4:     996,     3712,     4512,      2048,       256;
  5:    4192,    21040,    37888,     30208,     10240,      1024;
  6:   17016,   108032,   261504,    307200,    181760,     49152,     4096;
  7:   67328,   518528,  1588224,   2501120,   2170880,   1019904,   229376,    16384;
  8:  261392,  2370560,  8819456,  17555456,  20372480,  13991936,  5447680,  1048576,   65536;
  9: 1000064, 10447168, 45846528, 110835712, 162340864, 148721664, 84410368, 28049408, 4718592, 262144;
%F Formula
The row polynomials satisfy (2*n^3+3*n+2*n*x-2*n*x^2)*P(n,x) = (4*n^3+12*n^2+16*n+8+(4*n^3+12*n^2+14*n+6)*x-(4*n+4)*x^2-(4*n+4)*x^3)*P(n-1,x) + (2*n^2+2*n+1+(8*n^2+8*n+2)*x+(4*n^2+4*n-8)*x^2-12*x^3-4*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[((4*n^3 + 12*n^2 + 16*n + 8 + (4*n^3 + 12*n^2 + 14*n + 6)*x - (4*n + 4)*x^2 - (4*n + 4)*x^3)*P[n - 1, x] + (2*n^2 + 2*n + 1 + (8*n^2 + 8*n + 2)*x + (4*n^2 + 4*n - 8)*x^2 - 12*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 + 3*n + 2*n*x - 2*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] = ((4*n^3 + 12*n^2 + 16*n + 8 + (4*n^3 + 12*n^2 + 14*n + 6)*x - (4*n + 4)*x^2 - (4*n + 4)*x^3)*v[i - 1] + (2*n^2 + 2*n + 1 + (8*n^2 + 8*n + 2)*x + (4*n^2 + 4*n - 8)*x^2 - 12*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 + 3*n + 2*n*x - 2*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