Real-rooted atlas

A078812 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-52 + 16\,x^{2} + 84\,n - 16\,n x^{2} - 48\,n^{2} + 16\,n^{3}\right)\,P_{n}(x) = \left(34\,n - 12\,n x - 40\,n x^{2} - 16\,n x^{3} + 24\,n^{3} + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(5 - 28\,x - 112\,x^{2} - 80\,x^{3} - 16\,x^{4} - 20\,n - 48\,n x - 16\,n x^{2} + 20\,n^{2} + 48\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2527
32020444
471122608261
5235588492160161491
6744247230001648400328296
722859516153241216049609606445269
8686534462696127316042960139202240128243347

All 15 rows

Technical details
Node id
n-4073075195c11dd3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-353fe824af633f29

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:     5,      2;
   3:    20,     20,       4;
   4:    71,    122,      60,       8;
   5:   235,    588,     492,     160,      16;
   6:   744,   2472,    3000,    1648,     400,      32;
   7:  2285,   9516,   15324,   12160,    4960,     960,     64;
   8:  6865,  34462,   69612,   73160,   42960,   13920,   2240,   128;
   9: 20284, 119356,  291020,  383424,  297600,  138432,  37184,  5120,   256;
  10: 59155, 399574, 1143816, 1819280, 1768368, 1085280, 417536, 95744, 11520, 512;
%F Formula
The row polynomials satisfy (16*n^3-48*n^2+84*n-52-(16*n-16)*x^2)*P(n,x) = (24*n^3+34*n+(16*n^3-12*n)*x-40*n*x^2-16*n*x^3)*P(n-1,x) + (20*n^2-20*n+5+(48*n^2-48*n-28)*x+(16*n^2-16*n-112)*x^2-80*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((24*n^3 + 34*n + (16*n^3 - 12*n)*x - 40*n*x^2 - 16*n*x^3)*P[n - 1, x] + (20*n^2 - 20*n + 5 + (48*n^2 - 48*n - 28)*x + (16*n^2 - 16*n - 112)*x^2 - 80*x^3 - 16*x^4)*D[P[n - 1, x], x])/(16*n^3 - 48*n^2 + 84*n - 52 - (16*n - 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((24*n^3 + 34*n + (16*n^3 - 12*n)*x - 40*n*x^2 - 16*n*x^3)*v[i - 1] + (20*n^2 - 20*n + 5 + (48*n^2 - 48*n - 28)*x + (16*n^2 - 16*n - 112)*x^2 - 80*x^3 - 16*x^4)*deriv(v[i - 1], x))/(16*n^3 - 48*n^2 + 84*n - 52 - (16*n - 16)*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