Real-rooted atlas

A033282 → x ↦ x+1 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(1 - n\right)\,P_{n-2}(x) + \left(5 + 6\,x + 10\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15611
2317545151
32157839453782321
4159777701411211340340238221
5124257566018360022194013365032076659351
699955731025222061535862753247695156370531274111762011
7824675704098525692975519438156283777545486441182432253127410215197301

All 15 rows

Technical details
Node id
n-23d52b9eb8e45fdd · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-bffe79c3825a54d3

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:        5,         6;
  2:       31,        75,         45;
  3:      215,       783,        945,        378;
  4:     1597,      7770,      14112,      11340,        3402;
  5:    12425,     75660,     183600,     221940,      133650,       32076;
  6:    99955,    731025,    2220615,    3586275,     3247695,     1563705,      312741;
  7:   824675,   7040985,   25692975,   51943815,    62837775,    45486441,    18243225,    3127410;
  8:  6939769,  67748340,  288653904,  701080380,  1061674614,  1026485460,   618810192,  212663880,   31899582;
  9: 59334605, 651899826, 3176353440, 9008511876, 16389049950, 19834830288, 15969124080, 8247491928, 2479467510, 330595668;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(n-1)*P(n-2,x) + (10*n+5+(12*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1)*P[n - 2, x] + (10*n + 5 + (12*n + 6)*x)*P[n - 1, x])/(n + 2)]];
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] = 5 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1)*v[i - 2] + (10*n + 5 + (12*n + 6)*x)*v[i - 1])/(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. [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