Real-rooted atlas

A038331 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(17 + 54\,x + 37\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 1258\,x + 1369\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0289125813692916
14913320796981950653157464
2835217271242373846344440418741618503056
31419857154513856725897014638717015930368569343957459165024
4241375693152082541715103735497716378081244879357073083614256572640924794911296
5410338673625163037140819468893148070622455322271354755420848475033305321442671949318771331338925209984
66975757441121460247208925241294908402752093077610957226061670190784642014962076185810162812910735290088351247945392172301961339136
711858787649723229272278532022313116013210270178373478833529111748710272975125570722210588547631830289876712496917325374093564499131299617397950773904305912313344

All 15 rows

Technical details
Node id
n-ad084a9c619f7faf · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2c090b76ecce90b2

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,            1258,             1369;
  1:           4913,           32079,            69819,             50653;
  2:          83521,          727124,          2373846,           3444404,            1874161;
  3:        1419857,        15451385,         67258970,         146387170,          159303685,           69343957;
  4:       24137569,       315208254,       1715103735,        4977163780,         8124487935,         7073083614,          2565726409;
  5:      410338673,      6251630371,      40819468893,      148070622455,       322271354755,       420848475033,        305321442671,         94931877133;
  6:     6975757441,    121460247208,     925241294908,     4027520930776,     10957226061670,     19078464201496,      20761858101628,      12910735290088,       3512479453921;
  7:   118587876497,   2322927227853,   20223131160132,   102701783734788,    335291117487102,    729751255707222,    1058854763183028,     987671249691732,     537409356449913,     129961739795077;
  8:  2015993900449,  43877514303890,  429741537152805,  2494186176416280,   9499914995467890,  24811542694045548,   45001327435278690,   55968037482531480,   45679795298242605,   22093495765163090,   4808584372417849;
  9: 34271896307633, 820509517482743, 8929074160841615, 58301601873730545, 253783443450356490, 773293080631086246, 1683049646079423006, 2616505752308346690, 2847373906923789045, 2065741854042748915, 899205277642137763, 177917621779460413;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (17+54*x+37*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1258*x + 1369*x^2.
%t Mathematica
P[0, x_] := 289 + 1258*x + 1369*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 54*x + 37*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 + 1258*x + 1369*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 54*x + 37*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.