Real-rooted atlas

A091042 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(6 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(8 + 7\,x - 20\,n - 17\,n x + 8\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 2\,x^{2} - 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
233
3201030
411211221245
5576864360361836
62816563236968805513079
71331233280299521164818207890090
861440184320211200115200302403360105605865
927852897484813578249574403590406854457121364002072
1012451844980736817152070819843477760963072140448912017126069995
1155050242477260846792704481689602935296010762752230630426611213860210167941494
1224117248120586240257753088307494912224215040102875136294694405051904473616202402531072057117
131048576005767168001376256000186777600015876096008773632003168256007321600010296000800800286003006791746500
14452984832271790899271628226561089994752010590289920685847347230005821448798699521680307201976832012972963931235142752015487
1519461570561265002086436490444800615472168966731726848050048925696257827799049208135680224448307236095488036095488202675252780406267634562754
Technical details
Node id
n-d9b313e03aa9500f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9d7293d396b320fd

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 -2)
  • 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:       0;
   2:       3;
   3:      20,      10;
   4:     112,     112,      21;
   5:     576,     864,     360,      36;
   6:    2816,    5632,    3696,     880,      55;
   7:   13312,   33280,   29952,   11648,    1820,     78;
   8:   61440,  184320,  211200,  115200,   30240,   3360,    105;
   9:  278528,  974848, 1357824,  957440,  359040,  68544,   5712,  136;
  10: 1245184, 4980736, 8171520, 7081984, 3477760, 963072, 140448, 9120, 171;
%F Formula
The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (8*n^2-20*n+8+(6*n^2-17*n+7)*x)*P(n-1,x) - ((4*n-2)*x+(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 20*n + 8 + (6*n^2 - 17*n + 7)*x)*P[n - 1, x] - ((4*n - 2)*x + (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 20*n + 8 + (6*n^2 - 17*n + 7)*x)*v[i - 1] - ((4*n - 2)*x + (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); 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

Same fingerprint ?

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