Real-rooted atlas

A038320 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(17 + 52\,x + 35\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 1190\,x + 1225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0289119012252704
14913303456247542875140608
2835216878202124150291550015006257311616
31419857146161756018425012390875012755312552521875380204032
4241375692981699701534698375421289750065052093755357231250183826562519770609664
5410338673591370440536525821325125333700625258039971875318755259375218753609375643392968751028071702528
669757574411148948284408279186167003409076657000877335904375014450238425000148752454375008750144375000225187539062553459728531456
711858787649721973635939151809593547930086931454753500268464786738750552721619756250758637517312500669386044687500344536934765625788156386718752779905883635712

All 15 rows

Technical details
Node id
n-a8b636d9f68aad83 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4bdf144c22480455

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            289,            1190,             1225;
  1:           4913,           30345,            62475,             42875;
  2:          83521,          687820,          2124150,           2915500,            1500625;
  3:        1419857,        14616175,         60184250,         123908750,          127553125,           52521875;
  4:       24137569,       298169970,       1534698375,        4212897500,         6505209375,         5357231250,          1838265625;
  5:      410338673,      5913704405,      36525821325,      125333700625,       258039971875,       318755259375,        218753609375,         64339296875;
  6:     6975757441,    114894828440,     827918616700,     3409076657000,      8773359043750,     14450238425000,      14875245437500,       8750144375000,       2251875390625;
  7:   118587876497,   2197363593915,   18095935479300,    86931454753500,    268464786738750,    552721619756250,     758637517312500,     669386044687500,     344536934765625,      78815638671875;
  8:  2015993900449,  41505756773950,  384538628935125,  2111192472585000,   7606502290931250,  18792535071712500,   32242094485781250,   37931875865625000,   29285639455078125,   13398658574218750,   2758547353515625;
  9: 34271896307633, 776157651672865, 7989858178985375, 49349124046674375, 203202275486306250, 585700676401706250, 1205854333768218750, 1773315196717968750, 1825471526033203125, 1252774576689453125, 515848355107421875, 96549157373046875;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (17+52*x+35*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1190*x + 1225*x^2.
%t Mathematica
P[0, x_] := 289 + 1190*x + 1225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 52*x + 35*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1190*x + 1225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 52*x + 35*x^2)*v[i - 1])/(1 + x)); 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

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.