Real-rooted atlas

A001263 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
2915529
327816314185
481378504252421257
52431620324027009901328925
67296561182252227513365386142965445
72187255159355515592513513563063150151430491825
865619622844906497297211351347567562882885834448623768209
9196833542942047032557247683587147429968400982412888722267461679629324405
1059049127939589557652985255055724760631547284511052020407140566865088179058786231153133
111771474546773378897751515591003435339604809475444351430162590137001007275502462229034321982080121841801065
1253144115943230155889360736144200198758934033568175523729797280279734796014245753504854108601059078241337220074290014810069497
131594323552698646263917923445154856109096570442181931408829092418784266680505521703792118675724094162294669520451980360521515802674440120029657805
1447829691897244372466417681156206453135727569948113364329878921001089809723001193601117889817245399387873585391527986851067803600519165705652035917459694845979470140661
Technical details
Node id
n-fe3068304e7fd353 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-cb4056e025ca2479

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:     3,      2;
  2:     9,     15,       5;
  3:    27,     81,      63,      14;
  4:    81,    378,     504,     252,      42;
  5:   243,   1620,    3240,    2700,     990,     132;
  6:   729,   6561,   18225,   22275,   13365,    3861,     429;
  7:  2187,  25515,   93555,  155925,  135135,   63063,   15015,    1430;
  8:  6561,  96228,  449064,  972972, 1135134,  756756,  288288,   58344,   4862;
  9: 19683, 354294, 2047032, 5572476, 8358714, 7429968, 4009824, 1288872, 226746, 16796;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (3*n+6+(2*n+4)*x)*P(n-1,x) + (6*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 6 + (2*n + 4)*x)*P[n - 1, x] + (6*x + 2*x^2)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = ((3*n + 6 + (2*n + 4)*x)*v[i - 1] + (6*x + 2*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 - 1, v[i])))};
T(9)

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