Real-rooted atlas

A207824 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(12 + 2\,x + 6\,n + n x\right)\,P_{n-1}(x) + \left(32 + 12\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
212214
3106363145
48284267241330
5605141761068120511420
64240836732126002140180694066
72887883010321296302952037502527752979
819257362347036121672834782059220600633685902890
912637733176326561069129236797047863451068489016432945544035
10818978761287243668938000835985460925848015786961784161288854010347016740
115253607029185092927186753053314064009974636020475504290329228080017730660112617376056
1233419361966433052790560043576029106866801003477680240969960411294244989960421740236507921219577124644
1321109664455443636798404253351844024599961360955947807026295174725250048727666401681264156098403075693613145406256485
1413254482856030196835516031614619464020140043940087103342440270075085566176670192105499970413458376812664740854568391561092141073550481990
Technical details
Node id
n-8234b49d509b1125 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6e324802284d9792

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:      106,       36,        3;
  4:      828,      426,       72,       4;
  5:     6051,     4176,     1068,     120,      5;
  6:    42408,    36732,    12600,    2140,    180,      6;
  7:   288788,   301032,   129630,   29520,   3750,    252,    7;
  8:  1925736,  2347036,  1216728,  347820,  59220,   6006,  336,   8;
  9: 12637733, 17632656, 10691292, 3679704, 786345, 106848, 9016, 432, 9;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (6*n+12+(n+2)*x)*P(n-1,x) + (32+12*x+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[((6*n + 12 + (n + 2)*x)*P[n - 1, x] + (32 + 12*x + x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = ((6*n + 12 + (n + 2)*x)*v[i - 1] + (32 + 12*x + x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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