Real-rooted atlas

A104684 → basis A013611 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(-50 - 160\,x - 128\,x^{2} + 25\,n + 80\,n x + 64\,n x^{2}\right)\,P_{n-2}(x) + \left(21 + 24\,x - 14\,n - 16\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 7 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
27815
36112864253
4595180017285124635
56145243203456020480409689601
665527320840605440542720225280327681792575
771290941621769806400118169607557120235929626214436677005
878636675333165615075244822856960019855360097026048238551042097152762049275

All 15 rows

Technical details
Node id
n-a72e2a86e9694248 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-2c6cd6b0f6303dd6

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:         7,          8;
   3:        61,        128,          64;
   4:       595,       1800,        1728,         512;
   5:      6145,      24320,       34560,       20480,        4096;
   6:     65527,     320840,      605440,      542720,      225280,       32768;
   7:    712909,    4162176,     9806400,    11816960,     7557120,     2359296,      262144;
   8:   7863667,   53331656,   150752448,   228569600,   198553600,    97026048,    23855104,     2097152;
   9:  87615745,  676981760,  2232810496,  4083236864,  4492902400,  3009413120,  1174405120,   234881024,   16777216;
  10: 983726695, 8530832520, 32162780160, 68896235520, 91866120192, 78405304320, 42278584320, 13589544960, 2264924160, 134217728;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = (25*n-50+(80*n-160)*x+(64*n-128)*x^2)*P(n-2,x) - (14*n-21+(16*n-24)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 7 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((25*n - 50 + (80*n - 160)*x + (64*n - 128)*x^2)*P[n - 2, x] - (14*n - 21 + (16*n - 24)*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] = 7 + 8*x; for(n = 3, N, my(i = n); v[i] = ((25*n - 50 + (80*n - 160)*x + (64*n - 128)*x^2)*v[i - 2] - (14*n - 21 + (16*n - 24)*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