Real-rooted atlas

A133214 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(8 + 4\,x - x^{2} + 4\,n + n x - n x^{2}\right)\,P_{n-1}(x) + \left(8\,x + 6\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 16 + 32\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
016321361
16424022863595
22561536249615363216145
310248960217602208010020168365527
4409649152165120243200178320640328989712909
5163842580481139712226688023721601358448403396486397863667
665536131072073400321880883226199040211742729936640251494426572987615745
726214464880644482662414313062425366118427024076817755852870541568155546281462563983726695

All 15 rows

Technical details
Node id
n-82c3c8329527823d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e88a2f797c4f541a

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:      16,        32,         13;
  1:      64,       240,        228,         63;
  2:     256,      1536,       2496,       1536,         321;
  3:    1024,      8960,      21760,      22080,       10020,        1683;
  4:    4096,     49152,     165120,     243200,      178320,       64032,        8989;
  5:   16384,    258048,    1139712,    2266880,     2372160,     1358448,      403396,       48639;
  6:   65536,   1310720,    7340032,   18808832,    26199040,    21174272,     9936640,     2514944,      265729;
  7:  262144,   6488064,   44826624,  143130624,   253661184,   270240768,   177558528,    70541568,    15554628,    1462563;
  8: 1048576,  31457280,  262471680, 1018429440,  2224373760,  2989228032,  2570695680,  1421383680,   489425040,   95600160,   8097453;
  9: 4194304, 149946368, 1485045760, 6866534400, 18047631360, 29660725248, 31972233216, 23014210560, 10977384000, 3334581360, 584585892, 45046719;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (4*n+8+(n+4)*x-(n+1)*x^2)*P(n-1,x) + (8*x+6*x^2+x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 16 + 32*x + 13*x^2.
%t Mathematica
P[0, x_] := 16 + 32*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 8 + (n + 4)*x - (n + 1)*x^2)*P[n - 1, x] + (8*x + 6*x^2 + x^3)*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] = 16 + 32*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 8 + (n + 4)*x - (n + 1)*x^2)*v[i - 1] + (8*x + 6*x^2 + x^3)*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