Real-rooted atlas

A114687 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-8 - 64\,x + 4\,n - 32\,n x + 28\,n^{2} + 96\,n^{2} x\right)\,P_{n-1}(x) + \left(-1 - 12\,x - 32\,x^{2} - 2\,n - 24\,n x - 64\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
188
256128184
3336153615363408
4193613312276481638459280
5110001011203225604096001638401008120
662184716160309504061440005406720157286416996968
7351008485990426557440722534401009254406606028814680064285687584

All 15 rows

Technical details
Node id
n-e538c564ed1ba333 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-df9719bb32ea0a0b

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:        8;
  2:       56,       128;
  3:      336,      1536,       1536;
  4:     1936,     13312,      27648,      16384;
  5:    11000,    101120,     322560,     409600,      163840;
  6:    62184,    716160,    3095040,    6144000,     5406720,     1572864;
  7:   351008,   4859904,   26557440,   72253440,   100925440,    66060288,    14680064;
  8:  1981472,  32055296,  212029440,  735379456,  1422131200,  1497366528,   763363328,  134217728;
  9: 11194344, 207226368, 1609961472, 6800080896, 16886661120, 24838668288, 20610809856, 8455716864, 1207959552;
%F Formula
The row polynomials satisfy (4*n^2+4*n-8)*P(n,x) = (28*n^2+4*n-8+(96*n^2-32*n-64)*x)*P(n-1,x) - (2*n+1+(24*n+12)*x+(64*n+32)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((28*n^2 + 4*n - 8 + (96*n^2 - 32*n - 64)*x)*P[n - 1, x] - (2*n + 1 + (24*n + 12)*x + (64*n + 32)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 4*n - 8)]];
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] = 8; for(n = 2, N, my(i = n + 1); v[i] = ((28*n^2 + 4*n - 8 + (96*n^2 - 32*n - 64)*x)*v[i - 1] - (2*n + 1 + (24*n + 12)*x + (64*n + 32)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 4*n - 8)); 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. [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