Real-rooted atlas

A116414 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(20 + 8\,x + 10\,n + 4\,n x\right)\,P_{n-1}(x) + \left(13 + 20\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
220828
313812024282
48201128480642492
5450485605712160016020536
6235805743252800230404800384162036
711953435580041916025120081120134408961241150
859210020848403004800230464010192002607363584020489304204
9288231011725200200210401878240010614240370944078489692160460868616294
101384180063900720126266400140387520969696004323916812472320224870423040010240499566872
1165753584339695200762971040982300800803974080437794560160898304394752006197760563200225283599646256
123095804401769788176445514160065249308806184713600398879308817885952005583974401191168001655808013516804915225717016136
1314467565069068177040252961594084155812960044823972960334738944001776746496067617792001833384960345804800431185923194880106496182421943802
146718602740458185466961403199033602556722804483094610480002629806278401615772928007274505216024023577600575353856097230848010990387274547202293761286160366652
Technical details
Node id
n-98a17a3331a0d8a8 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-41fecb1af1a0a4d2

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:     138,      120,       24;
  4:     820,     1128,      480,       64;
  5:    4504,     8560,     5712,     1600,      160;
  6:   23580,    57432,    52800,    23040,     4800,     384;
  7:  119534,   355800,   419160,   251200,    81120,   13440,    896;
  8:  592100,  2084840,  3004800,  2304640,  1019200,  260736,  35840,  2048;
  9: 2882310, 11725200, 20021040, 18782400, 10614240, 3709440, 784896, 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) + (13+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] + (13 + 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] + (13 + 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