Real-rooted atlas

A207824 → basis A013611 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2483280
34245761921192
4331268164608102415760
5242046681668352307205120195212
6169632587712806400547840184320245762320480
711551524816512829632075571203840000103219211468826811984
8770294437552576778705928904192060641280246005765505024524288303439200
9505509322821224966842426889420042248052172804376494081477181442831155223592963380176020
103275915042059589856572032051292122777609480683520646633881629231677448446279681415577601048576037186641200
112101442808146961486724599521952084840038400102140272640838676643844756753612818402508800464781312069206016046137344404996841976
121336774478410292884464035842788864074513579008010275611443209870129561606738644828163270220185601105566105602479882240033218887682013265924374199518320
1384438657820709818877440272214518016062975901081609788905543680107705035653128601679822848502425295257621302909337606394635878401290000138241570347417687241523246914665132828
1453017931424048314936825602023335645696051558512486400891938226585601106227550453761011985644257286914046060134435280327278592132799424102403584317980672656928669696732828794883758096384500187701986240
Technical details
Node id
n-6996daab397299a6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-140c8fcd1272a2a2

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:        4;
  2:       48,        32;
  3:      424,       576,       192;
  4:     3312,      6816,      4608,      1024;
  5:    24204,     66816,     68352,     30720,      5120;
  6:   169632,    587712,    806400,    547840,    184320,     24576;
  7:  1155152,   4816512,   8296320,   7557120,   3840000,   1032192,    114688;
  8:  7702944,  37552576,  77870592,  89041920,  60641280,  24600576,   5505024,   524288;
  9: 50550932, 282122496, 684242688, 942004224, 805217280, 437649408, 147718144, 28311552, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (3*n+6+(2*n+4)*x)*P(n-1,x) + (4+6*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 6 + (2*n + 4)*x)*P[n - 1, x] + (4 + 6*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((3*n + 6 + (2*n + 4)*x)*v[i - 1] + (4 + 6*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); 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