Real-rooted atlas

A225470 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(3\,x + 3\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
2186354135
316278312426482835
41944118262664926406972076545
5291602123826126038757726211081749602525985
6524880443523615486876286285592957115616192548367416098513415
71102248010573707643167006097288476313080796921049751252465778512881798404433103675

All 15 rows

Technical details
Node id
n-5a6ba95ea9bda757 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-774e45ab36bcae1f

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 0)
  • 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:          1;
  1:          3,           6;
  2:         18,          63,           54;
  3:        162,         783,         1242,           648;
  4:       1944,       11826,        26649,         26406,          9720;
  5:      29160,      212382,       612603,        875772,        621108,        174960;
  6:     524880,     4435236,     15486876,      28628559,      29571156,      16192548,       3674160;
  7:   11022480,   105737076,    431670060,     972884763,    1308079692,    1049751252,     465778512,      88179840;
  8:  264539520,  2835296784,  13214982492,   35004325932,   57661801209,   60512181732,   39521968092,   14692335984,   2380855680;
  9: 7142567040, 84489198768, 441863430804, 1341566274924, 2606998410603, 3363682943034, 2882458590444, 1582352114328, 505053182880, 71425670400;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3*n+(6*n+3)*x+(3*n+3)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + (6*n + 3)*x + (3*n + 3)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n + (6*n + 3)*x + (3*n + 3)*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

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.