Real-rooted atlas

A094816 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 18\,x - 9\,n + 6\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(4 + 6\,x - 9\,x^{2} - 6\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-1 + 6\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)
\(P_{3}(x) = 20 + 33\,x + 9\,x^{2}\)
\(P_{4}(x) = 85 + 231\,x + 162\,x^{2} + 27\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2538
32033962
48523116227505
541315181719702814433
62333104671597510395283524342248
715244786991473211289255548510935729437338
8113825652383141693315256088816852721604082421874905605
9958817596240414506686182196001298259953751601255338148716656159415881
109006373598290751592883902254612861881084879617624130097494552436253144119683774042832
119339998065553719718788130092927346750276448383016596027636421053871577920502342277018698855904910804432670
121060328333779896319123777485776401027995174187206831528546151139129963612243952156131784859625963482856495390177147160994194073
131308238388510021206697832210797151758082195259065922458520649858605621025844681283392641438098227827969383739179510386475705223205225314412552054571433
141743356184291383969498675465746558951788944897559051083528867049289376307920905153632564509210771828224161185274722012442525781417183554245151761834975996063159432342899587539872
1524957436783882045201892752771666329725657143883242974107186302700721683165457918163538104331519506214475704690124951578289707809137930423150626493610263147658791112758516968512566860034782969762467944206026
Technical details
Node id
n-c4b574860ab4cdb2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-84896736c1c13b32

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:       5,        3;
   3:      20,       33,         9;
   4:      85,      231,       162,        27;
   5:     413,     1518,      1719,       702,        81;
   6:    2333,    10467,     15975,     10395,      2835,      243;
   7:   15244,    78699,    147321,    128925,     55485,    10935,      729;
   8:  113825,   652383,   1416933,   1525608,    881685,   272160,    40824,    2187;
   9:  958817,  5962404,  14506686,  18219600,  12982599,  5375160,  1255338,  148716,   6561;
  10: 9006373, 59829075, 159288390, 225461286, 188108487, 96176241, 30097494, 5524362, 531441, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(6*n-18)*x)*P(n-3,x) - (n^2-4+(6*n-6)*x+9*x^2)*P(n-2,x) + (2*n-1+6*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 5 + 3*x, P(3,x) = 20 + 33*x + 9*x^2, P(4,x) = 85 + 231*x + 162*x^2 + 27*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 3*x;
P[3, x_] := 20 + 33*x + 9*x^2;
P[4, x_] := 85 + 231*x + 162*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (6*n - 18)*x)*P[n - 3, x] - (n^2 - 4 + (6*n - 6)*x + 9*x^2)*P[n - 2, x] + (2*n - 1 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; v[3] = 20 + 33*x + 9*x^2; v[4] = 85 + 231*x + 162*x^2 + 27*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (6*n - 18)*x)*v[i - 3] - (n^2 - 4 + (6*n - 6)*x + 9*x^2)*v[i - 2] + (2*n - 1 + 6*x)*v[i - 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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023