Real-rooted atlas

A063007 → basis A013609 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
244
3364884
42647204801464
51800744010080448023800
6118206552013440012096040320373020
7758525275201451520197568013305603548165715948
8479248400982413849920252806402572416013837824307507286256688

All 15 rows

Technical details
Node id
n-8aa50ef542759ee0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9a4b425d2a954ae8

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 -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        0;
   2:        4;
   3:       36,        48;
   4:      264,       720,        480;
   5:     1800,      7440,      10080,       4480;
   6:    11820,     65520,     134400,     120960,      40320;
   7:    75852,    527520,    1451520,    1975680,    1330560,     354816;
   8:   479248,   4009824,   13849920,   25280640,   25724160,   13837824,    3075072;
   9:  2994192,  29282400,  121746240,  278974080,  380540160,  309044736,  138378240,   26357760;
  10: 18548820, 207614880, 1009451520, 2784862080, 4768283520, 5189184000, 3505582080, 1344245760, 224040960;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (3*n+4*n*x)*P(n-1,x) + (2+6*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 4*n*x)*P[n - 1, x] + (2 + 6*x + 4*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((3*n + 4*n*x)*v[i - 1] + (2 + 6*x + 4*x^2)*deriv(v[i - 1], x))/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023