Real-rooted atlas

A103327 → D → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(3 - 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(4 + 4\,x - 12\,n - 13\,n x + 8\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 2\,x^{2} - 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
212214
380483131
444848012041052
523043584168024057813
61126423040161284480420655338
75324813516812672053760100806727379655
824576074547287859250688014784020160100882545720
911141123932160559104041000961647360354816369601440916777993
10498073620054016334233602981888015375360461260876876863360198010109099078
11220200969961472019051315220054016012673024049201152115315201537536102960264011701794187
1296468992484442112104595456012700876809525657604562288641394032642635776028828801601603432124474555476
1341943040023152558085571084288767033344066679603203810263040144472473635846553656010240512512024024043681328318897549
14181193932810905190400289406976004456867430444104417280293390254081333592064041277849608513556481120204808712704349440546014178106093666
Technical details
Node id
n-843b3e5869e910c1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-83163cb94ac570df

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:       1;
  2:      12,       2;
  3:      80,      48,       3;
  4:     448,     480,     120,       4;
  5:    2304,    3584,    1680,     240,       5;
  6:   11264,   23040,   16128,    4480,     420,      6;
  7:   53248,  135168,  126720,   53760,   10080,    672,     7;
  8:  245760,  745472,  878592,  506880,  147840,  20160,  1008,    8;
  9: 1114112, 3932160, 5591040, 4100096, 1647360, 354816, 36960, 1440, 9;
%F Formula
The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (8*n^2-12*n+4+(6*n^2-13*n+4)*x)*P(n-1,x) - ((4*n-2)*x+(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 12*n + 4 + (6*n^2 - 13*n + 4)*x)*P[n - 1, x] - ((4*n - 2)*x + (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 12*n + 4 + (6*n^2 - 13*n + 4)*x)*v[i - 1] - ((4*n - 2)*x + (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3)); 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