Real-rooted atlas

A038319 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 25 + 110\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
025110121256
1125825181513314096
2625550018150266201464165536
33125343751512503327503660251610511048576
4156252062501134375332750054903754831530177156116777216
578125120312579406252911562564054375845517756200463519487171268435456
6390625687500052937500232925000640543750112735700012400927007794868402143588814294967296
7195312538671875340312500174693750057648937501268276625018601390500175384539009646149645235794769168719476736

All 15 rows

Technical details
Node id
n-d46be0acd5f53980 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-2235b7dd30e0ad6a

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:       25,        110,         121;
  1:      125,        825,        1815,        1331;
  2:      625,       5500,       18150,       26620,        14641;
  3:     3125,      34375,      151250,      332750,       366025,        161051;
  4:    15625,     206250,     1134375,     3327500,      5490375,       4831530,       1771561;
  5:    78125,    1203125,     7940625,    29115625,     64054375,      84551775,      62004635,      19487171;
  6:   390625,    6875000,    52937500,   232925000,    640543750,    1127357000,    1240092700,     779486840,     214358881;
  7:  1953125,   38671875,   340312500,  1746937500,   5764893750,   12682766250,   18601390500,   17538453900,    9646149645,    2357947691;
  8:  9765625,  214843750,  2126953125, 12478125000,  48040781250,  126827662500,  232517381250,  292307565000,  241153741125,  117897384550,   25937424601;
  9: 48828125, 1181640625, 12998046875, 85787109375, 377463281250, 1162586906250, 2557691193750, 4019229018750, 4421151920625, 3242178075125, 1426558353055, 285311670611;
%F Formula
The row polynomials satisfy P(n,x) = (5+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 110*x + 121*x^2.
%t Mathematica
P[0, x_] := 25 + 110*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 11*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 25 + 110*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 11*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.