Real-rooted atlas

A207607 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x + 15\,n + 12\,n x - 15\,n^{2} - 12\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-6\,n - 3\,n x - 21\,n^{2} - 16\,n^{2} x - 2\,n^{2} x^{2} + 3\,n^{3} + 7\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-15 - 18\,x - 3\,x^{2} - 20\,n - 34\,n x - 16\,n x^{2} - 2\,n x^{3} + 5\,n^{2} + 16\,n^{2} x + 13\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
29413
33330669
4111152638334
5355648411108101532
6109825022142864165126783
7331690549768534015652341429291
8983731292407432818011205256831516124156
9287791044961593991335246727520916392740818518742
1083265339754594198584924357525140910358685696513202142673
112387271081296213282624135121736455830844268072576727929630228767985
1267928433815887426224949938078687304435368173766647368888155106807592435601546
13192051810422036252182283598626033743960219177361011952233492967894351293601400390026143611280
145400081317255568386151413208370413834597510181059254191592211393686048108125455018354617952105328576063619
Technical details
Node id
n-db79a38d51017217 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-0b4f8b4cb940af7f

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     2;
  2:     9,      4;
  3:    33,     30,      6;
  4:   111,    152,     63,      8;
  5:   355,    648,    411,    108,    10;
  6:  1098,   2502,   2142,    864,   165,    12;
  7:  3316,   9054,   9768,   5340,  1565,   234,   14;
  8:  9837,  31292,  40743,  28180, 11205,  2568,  315,  16;
  9: 28779, 104496, 159399, 133524, 67275, 20916, 3927, 408, 18;
%F Formula
The row polynomials satisfy (2*n^3-15*n^2+15*n-2+(4*n^3-12*n^2+12*n-4)*x)*P(n,x) = (3*n^3-21*n^2-6*n+(7*n^3-16*n^2-3*n)*x+(2*n^3-2*n^2)*x^2)*P(n-1,x) + (5*n^2-20*n-15+(16*n^2-34*n-18)*x+(13*n^2-16*n-3)*x^2+(2*n^2-2*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 21*n^2 - 6*n + (7*n^3 - 16*n^2 - 3*n)*x + (2*n^3 - 2*n^2)*x^2)*P[n - 1, x] + (5*n^2 - 20*n - 15 + (16*n^2 - 34*n - 18)*x + (13*n^2 - 16*n - 3)*x^2 + (2*n^2 - 2*n)*x^3)*D[P[n - 1, x], x])/(2*n^3 - 15*n^2 + 15*n - 2 + (4*n^3 - 12*n^2 + 12*n - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^3 - 21*n^2 - 6*n + (7*n^3 - 16*n^2 - 3*n)*x + (2*n^3 - 2*n^2)*x^2)*v[i - 1] + (5*n^2 - 20*n - 15 + (16*n^2 - 34*n - 18)*x + (13*n^2 - 16*n - 3)*x^2 + (2*n^2 - 2*n)*x^3)*deriv(v[i - 1], x))/(2*n^3 - 15*n^2 + 15*n - 2 + (4*n^3 - 12*n^2 + 12*n - 4)*x)); 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