Real-rooted atlas

A367270 → basis A013611 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-32 - 192\,x - 288\,x^{2} + 80\,n + 480\,n x + 720\,n x^{2} - 32\,n^{2} - 192\,n^{2} x - 288\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(22 + 48\,x - 48\,n - 96\,n x + 24\,n^{2} + 48\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2112435
3106432432970
41011590411232691225059
5972673440203040241920103680631806
694526873360315360055296004665600149299215809678
792683610118304450273601040256001306368008360755220901888395244340

All 15 rows

Technical details
Node id
n-38d919b9fc1f2f68 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f3afab2bbcf25425

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:        1;
  1:        1;
  2:       11,         24;
  3:      106,        432,        432;
  4:     1011,       5904,      11232,        6912;
  5:     9726,      73440,     203040,      241920,      103680;
  6:    94526,     873360,    3153600,     5529600,     4665600,     1492992;
  7:   926836,   10118304,   45027360,   104025600,   130636800,    83607552,    20901888;
  8:  9153731,  115235232,  609118272,  1745210880,  2910297600,  2800852992,  1421328384,   286654464;
  9: 90941686, 1296575424, 7934697792, 27129779712, 56422033920, 72613158912, 55933452288, 23219011584, 3869835264;
%F Formula
The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(32*n^2-80*n+32+(192*n^2-480*n+192)*x+(288*n^2-720*n+288)*x^2)*P(n-2,x) + (24*n^2-48*n+22+(48*n^2-96*n+48)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(32*n^2 - 80*n + 32 + (192*n^2 - 480*n + 192)*x + (288*n^2 - 720*n + 288)*x^2)*P[n - 2, x] + (24*n^2 - 48*n + 22 + (48*n^2 - 96*n + 48)*x)*P[n - 1, x])/(2*n^2 - 3*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(32*n^2 - 80*n + 32 + (192*n^2 - 480*n + 192)*x + (288*n^2 - 720*n + 288)*x^2)*v[i - 2] + (24*n^2 - 48*n + 22 + (48*n^2 - 96*n + 48)*x)*v[i - 1])/(2*n^2 - 3*n)); 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