Real-rooted atlas

A114656 → basis A007318 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(8\,x^{2} - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-4 - 2\,x + 8\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
32020444
4112168728360
56721344896224163152
6422410560960038406403228896
72745682368950405280014400172864273856
81830406406408968966406402464004928044801282661504

All 15 rows

Technical details
Node id
n-ca7ba1032f4945db · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-1e5fbd8fe1b88630

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       4,        2;
   3:      20,       20,        4;
   4:     112,      168,       72,        8;
   5:     672,     1344,      896,      224,       16;
   6:    4224,    10560,     9600,     3840,      640,       32;
   7:   27456,    82368,    95040,    52800,    14400,     1728,      64;
   8:  183040,   640640,   896896,   640640,   246400,    49280,    4480,    128;
   9: 1244672,  4978688,  8200192,  7175168,  3587584,  1025024,  157696,  11264,   256;
  10: 8599552, 38697984, 73322496, 76038144, 46964736, 17611776, 3913728, 479232, 27648, 512;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(4*n-8)*x^2*P(n-2,x) + (8*n-4+(4*n-2)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8)*x^2*P[n - 2, x] + (8*n - 4 + (4*n - 2)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(4*n - 8)*x^2*v[i - 2] + (8*n - 4 + (4*n - 2)*x)*v[i - 1])/(n + 1)); 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