Real-rooted atlas

A207823 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(20 + 8\,x + 10\,n + 4\,n x\right)\,P_{n-1}(x) + \left(21 + 20\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
220828
314612024290
49401176480642660
5565695205904160016022840
6326206904857600236804800384188132
718274246620048924027040083040134408961505958
81002300299292037872002632960108640026611235840204811805780
9540951018507600274255202299360011931360392448079923292160460891088070
10288278001111831201888632001854697601160544004799155213117440228556823040010240694033480
11152061488652719200125043984014079904001032015040514563840176768256413184006289920563200225285234752112
1279535708037606588328021493600101889884808563996800500166374420713728006083481601241856001678336013516804915239154249288
13413077911421332003280501390046567092760160067260440160450473472002185064448077349888001983421440359321600436592643194880106496290812512970
142132493706011942247402430666953952047815303193650505720800038166012864021229232640088010926080271966464006187427840100745216011118182474547202293762147100963980
Technical details
Node id
n-232e9f1eefacb65f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c9133ad343d619e1

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:      20,        8;
  3:     146,      120,       24;
  4:     940,     1176,      480,       64;
  5:    5656,     9520,     5904,     1600,      160;
  6:   32620,    69048,    57600,    23680,     4800,     384;
  7:  182742,   466200,   489240,   270400,    83040,   13440,    896;
  8: 1002300,  2992920,  3787200,  2632960,  1086400,  266112,  35840,  2048;
  9: 5409510, 18507600, 27425520, 22993600, 11931360, 3924480, 799232, 92160, 4608;
%F Formula
The row polynomials satisfy (4*n-4)*P(n,x) = (10*n+20+(4*n+8)*x)*P(n-1,x) + (21+20*x+4*x^2)*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[((10*n + 20 + (4*n + 8)*x)*P[n - 1, x] + (21 + 20*x + 4*x^2)*D[P[n - 1, x], x])/(4*n - 4)]];
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] = ((10*n + 20 + (4*n + 8)*x)*v[i - 1] + (21 + 20*x + 4*x^2)*deriv(v[i - 1], x))/(4*n - 4)); 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